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IB · MATH AA HL

Mathematics AA: Higher Level

Functions — Topic 2

Name: ____________________Date: October 10, 2026
  1. 1.

    Show that (x − 2) is a factor of p(x) = x³ − 3x² − 4x + 12, and hence fully factorise p(x).

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Keep each expression equivalent to the previous one. Expand brackets with their signs intact, combine only like terms, or take out a common factor as requested. Check an algebraic result by expanding it back or substituting a permitted value.
    2. Work through this mathematical step: Applies the factor theorem by evaluating p(2) = 8 − 12 − 8 + 12 = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Concludes (x − 2) is a factor because p(2) = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Divides p(x) by (x − 2) to obtain the quotient x² − x − 6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Factorises the quotient to (x − 3)(x + 2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Work through this mathematical step: States the complete factorisation p(x) = (x − 2)(x − 3)(x + 2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Polynomial division (or synthetic/coefficient matching) must be shown; simply stating the quotient without working loses method marks.

    Marking points

    • Applies the factor theorem by evaluating p(2) = 8 − 12 − 8 + 12 = 0.
    • Concludes (x − 2) is a factor because p(2) = 0.
    • Divides p(x) by (x − 2) to obtain the quotient x² − x − 6.
    • Factorises the quotient to (x − 3)(x + 2).
    • States the complete factorisation p(x) = (x − 2)(x − 3)(x + 2).

    Examiner tip: Polynomial division (or synthetic/coefficient matching) must be shown; simply stating the quotient without working loses method marks.

  2. 2.

    Marking analysis: A learner attempts the following task: “Show that (x − 2) is a factor of p(x) = x³ − 3x² − 4x + 12, and hence fully factorise p(x).” Their response addresses only this point: “Applies the factor theorem by evaluating p(2) = 8 − 12 − 8 + 12 = 0.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: Applies the factor theorem by evaluating p(2) = 8 − 12 − 8 + 12 = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Concludes (x − 2) is a factor because p(2) = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Divides p(x) by (x − 2) to obtain the quotient x² − x − 6. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Factorises the quotient to (x − 3)(x + 2). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: States the complete factorisation p(x) = (x − 2)(x − 3)(x + 2). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: Applies the factor theorem by evaluating p(2) = 8 − 12 − 8 + 12 = 0.
    • Identifies the missing requirement: Concludes (x − 2) is a factor because p(2) = 0.
    • Identifies the missing requirement: Divides p(x) by (x − 2) to obtain the quotient x² − x − 6.
    • Identifies the missing requirement: Factorises the quotient to (x − 3)(x + 2).
    • Identifies the missing requirement: States the complete factorisation p(x) = (x − 2)(x − 3)(x + 2).

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  3. 3.

    The rational function f(x) = (2x + 1)/(x − 3) has a vertical and a horizontal asymptote. State the equation of each asymptote.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
    2. Work through this mathematical step: Identifies the vertical asymptote where the denominator is zero: x = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Identifies the horizontal asymptote from the ratio of leading coefficients as x → ±∞. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the horizontal asymptote y = 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For a linear-over-linear rational function, the horizontal asymptote is simply the ratio of the x-coefficients.

    Marking points

    • Identifies the vertical asymptote where the denominator is zero: x = 3.
    • Identifies the horizontal asymptote from the ratio of leading coefficients as x → ±∞.
    • States the horizontal asymptote y = 2.

    Examiner tip: For a linear-over-linear rational function, the horizontal asymptote is simply the ratio of the x-coefficients.

  4. 4.

    Marking analysis: A learner attempts the following task: “The rational function f(x) = (2x + 1)/(x − 3) has a vertical and a horizontal asymptote. State the equation of each asymptote.” Their response addresses only this point: “Identifies the vertical asymptote where the denominator is zero: x = 3.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: Identifies the vertical asymptote where the denominator is zero: x = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Identifies the horizontal asymptote from the ratio of leading coefficients as x → ±∞. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States the horizontal asymptote y = 2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: Identifies the vertical asymptote where the denominator is zero: x = 3.
    • Identifies the missing requirement: Identifies the horizontal asymptote from the ratio of leading coefficients as x → ±∞.
    • Identifies the missing requirement: States the horizontal asymptote y = 2.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  5. 5.

    The cubic equation 2x³ − 5x² + 3x − 1 = 0 has roots α, β, γ. Find α + β + γ and αβγ.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the relations for ax³ + bx² + cx + d = 0: sum of roots = −b/a, product of roots = −d/a. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Identifies a = 2, b = −5, d = −1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Calculates α + β + γ = −(−5)/2 = 5/2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Calculates αβγ = −(−1)/2 = 1/2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The sum-and-product-of-roots relations let you answer questions about roots without ever solving the cubic — always check whether the question actually requires the individual roots first.

    Marking points

    • States the relations for ax³ + bx² + cx + d = 0: sum of roots = −b/a, product of roots = −d/a.
    • Identifies a = 2, b = −5, d = −1.
    • Calculates α + β + γ = −(−5)/2 = 5/2.
    • Calculates αβγ = −(−1)/2 = 1/2.

    Examiner tip: The sum-and-product-of-roots relations let you answer questions about roots without ever solving the cubic — always check whether the question actually requires the individual roots first.

  6. 6.

    Marking analysis: A learner attempts the following task: “The cubic equation 2x³ − 5x² + 3x − 1 = 0 has roots α, β, γ. Find α + β + γ and αβγ.” Their response addresses only this point: “States the relations for ax³ + bx² + cx + d = 0: sum of roots = −b/a, product of roots = −d/a.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: States the relations for ax³ + bx² + cx + d = 0: sum of roots = −b/a, product of roots = −d/a. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Identifies a = 2, b = −5, d = −1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates α + β + γ = −(−5)/2 = 5/2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Calculates αβγ = −(−1)/2 = 1/2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: States the relations for ax³ + bx² + cx + d = 0: sum of roots = −b/a, product of roots = −d/a.
    • Identifies the missing requirement: Identifies a = 2, b = −5, d = −1.
    • Identifies the missing requirement: Calculates α + β + γ = −(−5)/2 = 5/2.
    • Identifies the missing requirement: Calculates αβγ = −(−1)/2 = 1/2.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  7. 7.

    The function f(x) = √(x + 4) has domain x ≥ −4. Find f⁻¹(x) and state its domain and range.

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Writes y = √(x + 4) and swaps x and y: x = √(y + 4). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Squares both sides: x² = y + 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Rearranges to obtain f⁻¹(x) = x² − 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States the domain of f⁻¹ as x ≥ 0, matching the range of f. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Work through this mathematical step: States the range of f⁻¹ as y ≥ −4, matching the domain of f. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The domain of f⁻¹ always equals the range of f, and the range of f⁻¹ always equals the domain of f — stating only the algebraic inverse without these restrictions loses marks.

    Marking points

    • Writes y = √(x + 4) and swaps x and y: x = √(y + 4).
    • Squares both sides: x² = y + 4.
    • Rearranges to obtain f⁻¹(x) = x² − 4.
    • States the domain of f⁻¹ as x ≥ 0, matching the range of f.
    • States the range of f⁻¹ as y ≥ −4, matching the domain of f.

    Examiner tip: The domain of f⁻¹ always equals the range of f, and the range of f⁻¹ always equals the domain of f — stating only the algebraic inverse without these restrictions loses marks.

  8. 8.

    Marking analysis: A learner attempts the following task: “The function f(x) = √(x + 4) has domain x ≥ −4. Find f⁻¹(x) and state its domain and range.” Their response addresses only this point: “Writes y = √(x + 4) and swaps x and y: x = √(y + 4).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: Writes y = √(x + 4) and swaps x and y: x = √(y + 4). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Squares both sides: x² = y + 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Rearranges to obtain f⁻¹(x) = x² − 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: States the domain of f⁻¹ as x ≥ 0, matching the range of f. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: States the range of f⁻¹ as y ≥ −4, matching the domain of f. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: Writes y = √(x + 4) and swaps x and y: x = √(y + 4).
    • Identifies the missing requirement: Squares both sides: x² = y + 4.
    • Identifies the missing requirement: Rearranges to obtain f⁻¹(x) = x² − 4.
    • Identifies the missing requirement: States the domain of f⁻¹ as x ≥ 0, matching the range of f.
    • Identifies the missing requirement: States the range of f⁻¹ as y ≥ −4, matching the domain of f.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  9. 9.

    Solve the inequality (x − 1)(x + 2)(x − 3) ≥ 0 using a sign diagram.

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Identifies the critical values x = −2, 1, 3 where the expression equals zero. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Tests the sign of the expression in each of the four intervals created by these values. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Constructs a correct sign diagram: negative, positive, negative, positive from left to right. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Selects the intervals where the expression is positive or zero, including the critical values since the inequality is not strict. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Work through this mathematical step: States the solution −2 ≤ x ≤ 1 or x ≥ 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The sign of a product only changes at a critical value if that factor has odd multiplicity — test one point per interval rather than assuming signs simply alternate.

    Marking points

    • Identifies the critical values x = −2, 1, 3 where the expression equals zero.
    • Tests the sign of the expression in each of the four intervals created by these values.
    • Constructs a correct sign diagram: negative, positive, negative, positive from left to right.
    • Selects the intervals where the expression is positive or zero, including the critical values since the inequality is not strict.
    • States the solution −2 ≤ x ≤ 1 or x ≥ 3.

    Examiner tip: The sign of a product only changes at a critical value if that factor has odd multiplicity — test one point per interval rather than assuming signs simply alternate.

  10. 10.

    Marking analysis: A learner attempts the following task: “Solve the inequality (x − 1)(x + 2)(x − 3) ≥ 0 using a sign diagram.” Their response addresses only this point: “Identifies the critical values x = −2, 1, 3 where the expression equals zero.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: Identifies the critical values x = −2, 1, 3 where the expression equals zero. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Tests the sign of the expression in each of the four intervals created by these values. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Constructs a correct sign diagram: negative, positive, negative, positive from left to right. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Selects the intervals where the expression is positive or zero, including the critical values since the inequality is not strict. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: States the solution −2 ≤ x ≤ 1 or x ≥ 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: Identifies the critical values x = −2, 1, 3 where the expression equals zero.
    • Identifies the missing requirement: Tests the sign of the expression in each of the four intervals created by these values.
    • Identifies the missing requirement: Constructs a correct sign diagram: negative, positive, negative, positive from left to right.
    • Identifies the missing requirement: Selects the intervals where the expression is positive or zero, including the critical values since the inequality is not strict.
    • Identifies the missing requirement: States the solution −2 ≤ x ≤ 1 or x ≥ 3.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  11. 11.

    Show algebraically that f(x) = (x + 1)/(x − 1), x ≠ 1, is a self-inverse function.

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States that f is self-inverse if f(f(x)) = x for all x in the domain. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes f(x) into itself: f(f(x)) = [((x+1)/(x−1)) + 1] / [((x+1)/(x−1)) − 1]. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Combines each part of the compound fraction over the common denominator (x − 1): numerator becomes 2x/(x−1), denominator becomes 2/(x−1). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Simplifies the compound fraction to obtain f(f(x)) = 2x/2 = x. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Work through this mathematical step: Concludes that since f(f(x)) = x, f is self-inverse. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A self-inverse function's graph is always symmetric about the line y = x — this can be used as a quick visual check before attempting the algebraic proof.

    Marking points

    • States that f is self-inverse if f(f(x)) = x for all x in the domain.
    • Substitutes f(x) into itself: f(f(x)) = [((x+1)/(x−1)) + 1] / [((x+1)/(x−1)) − 1].
    • Combines each part of the compound fraction over the common denominator (x − 1): numerator becomes 2x/(x−1), denominator becomes 2/(x−1).
    • Simplifies the compound fraction to obtain f(f(x)) = 2x/2 = x.
    • Concludes that since f(f(x)) = x, f is self-inverse.

    Examiner tip: A self-inverse function's graph is always symmetric about the line y = x — this can be used as a quick visual check before attempting the algebraic proof.

  12. 12.

    Marking analysis: A learner attempts the following task: “Show algebraically that f(x) = (x + 1)/(x − 1), x ≠ 1, is a self-inverse function.” Their response addresses only this point: “States that f is self-inverse if f(f(x)) = x for all x in the domain.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: States that f is self-inverse if f(f(x)) = x for all x in the domain. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes f(x) into itself: f(f(x)) = [((x+1)/(x−1)) + 1] / [((x+1)/(x−1)) − 1]. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Combines each part of the compound fraction over the common denominator (x − 1): numerator becomes 2x/(x−1), denominator becomes 2/(x−1). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Simplifies the compound fraction to obtain f(f(x)) = 2x/2 = x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Concludes that since f(f(x)) = x, f is self-inverse. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: States that f is self-inverse if f(f(x)) = x for all x in the domain.
    • Identifies the missing requirement: Substitutes f(x) into itself: f(f(x)) = [((x+1)/(x−1)) + 1] / [((x+1)/(x−1)) − 1].
    • Identifies the missing requirement: Combines each part of the compound fraction over the common denominator (x − 1): numerator becomes 2x/(x−1), denominator becomes 2/(x−1).
    • Identifies the missing requirement: Simplifies the compound fraction to obtain f(f(x)) = 2x/2 = x.
    • Identifies the missing requirement: Concludes that since f(f(x)) = x, f is self-inverse.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  13. 13.

    Given f(x) = √x and g(x) = x − 4, find the domain of the composite function (f ∘ g)(x).

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Forms the composite (f ∘ g)(x) = f(g(x)) = √(x − 4). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Recognises that the domain of f restricts its input to values ≥ 0, so g(x) must satisfy g(x) ≥ 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Sets up and solves the inequality x − 4 ≥ 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States the domain of the composite function as x ≥ 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The domain of a composite function f(g(x)) is restricted by the domain of the outer function applied to g(x), not by the domain of g alone — always check what values g(x) is allowed to output.

    Marking points

    • Forms the composite (f ∘ g)(x) = f(g(x)) = √(x − 4).
    • Recognises that the domain of f restricts its input to values ≥ 0, so g(x) must satisfy g(x) ≥ 0.
    • Sets up and solves the inequality x − 4 ≥ 0.
    • States the domain of the composite function as x ≥ 4.

    Examiner tip: The domain of a composite function f(g(x)) is restricted by the domain of the outer function applied to g(x), not by the domain of g alone — always check what values g(x) is allowed to output.

  14. 14.

    Marking analysis: A learner attempts the following task: “Given f(x) = √x and g(x) = x − 4, find the domain of the composite function (f ∘ g)(x).” Their response addresses only this point: “Forms the composite (f ∘ g)(x) = f(g(x)) = √(x − 4).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: Forms the composite (f ∘ g)(x) = f(g(x)) = √(x − 4). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Recognises that the domain of f restricts its input to values ≥ 0, so g(x) must satisfy g(x) ≥ 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Sets up and solves the inequality x − 4 ≥ 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: States the domain of the composite function as x ≥ 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: Forms the composite (f ∘ g)(x) = f(g(x)) = √(x − 4).
    • Identifies the missing requirement: Recognises that the domain of f restricts its input to values ≥ 0, so g(x) must satisfy g(x) ≥ 0.
    • Identifies the missing requirement: Sets up and solves the inequality x − 4 ≥ 0.
    • Identifies the missing requirement: States the domain of the composite function as x ≥ 4.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  15. 15.

    Determine algebraically whether f(x) = x³ − 2x is odd, even, or neither.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the tests: f is even if f(−x) = f(x) for all x; f is odd if f(−x) = −f(x) for all x. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Calculates f(−x) = (−x)³ − 2(−x) = −x³ + 2x. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Recognises that −x³ + 2x = −(x³ − 2x) = −f(x). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Concludes that since f(−x) = −f(x), the function is odd. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Every term of an odd function has an odd power of x, and every term of an even function has an even power (including the constant term, x⁰) — a quick glance at the powers often predicts the answer before any substitution.

    Marking points

    • States the tests: f is even if f(−x) = f(x) for all x; f is odd if f(−x) = −f(x) for all x.
    • Calculates f(−x) = (−x)³ − 2(−x) = −x³ + 2x.
    • Recognises that −x³ + 2x = −(x³ − 2x) = −f(x).
    • Concludes that since f(−x) = −f(x), the function is odd.

    Examiner tip: Every term of an odd function has an odd power of x, and every term of an even function has an even power (including the constant term, x⁰) — a quick glance at the powers often predicts the answer before any substitution.

  16. 16.

    Marking analysis: A learner attempts the following task: “Determine algebraically whether f(x) = x³ − 2x is odd, even, or neither.” Their response addresses only this point: “States the tests: f is even if f(−x) = f(x) for all x; f is odd if f(−x) = −f(x) for all x.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: States the tests: f is even if f(−x) = f(x) for all x; f is odd if f(−x) = −f(x) for all x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Calculates f(−x) = (−x)³ − 2(−x) = −x³ + 2x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Recognises that −x³ + 2x = −(x³ − 2x) = −f(x). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Concludes that since f(−x) = −f(x), the function is odd. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: States the tests: f is even if f(−x) = f(x) for all x; f is odd if f(−x) = −f(x) for all x.
    • Identifies the missing requirement: Calculates f(−x) = (−x)³ − 2(−x) = −x³ + 2x.
    • Identifies the missing requirement: Recognises that −x³ + 2x = −(x³ − 2x) = −f(x).
    • Identifies the missing requirement: Concludes that since f(−x) = −f(x), the function is odd.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  17. 17.

    Express (5x − 3)/((x − 1)(x + 2)) in the form A/(x − 1) + B/(x + 2), finding the values of A and B.

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
    2. Work through this mathematical step: States 5x − 3 = A(x + 2) + B(x − 1). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes x = 1 to find A: 5(1) − 3 = A(3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Obtains A = 2/3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Substitutes x = −2 to find B: 5(−2) − 3 = B(−3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Work through this mathematical step: Obtains B = 13/3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Substituting the value that makes one bracket zero eliminates the other unknown entirely — this 'cover-up' technique is far faster than expanding and comparing coefficients.

    Marking points

    • States 5x − 3 = A(x + 2) + B(x − 1).
    • Substitutes x = 1 to find A: 5(1) − 3 = A(3).
    • Obtains A = 2/3.
    • Substitutes x = −2 to find B: 5(−2) − 3 = B(−3).
    • Obtains B = 13/3.

    Examiner tip: Substituting the value that makes one bracket zero eliminates the other unknown entirely — this 'cover-up' technique is far faster than expanding and comparing coefficients.

  18. 18.

    Marking analysis: A learner attempts the following task: “Express (5x − 3)/((x − 1)(x + 2)) in the form A/(x − 1) + B/(x + 2), finding the values of A and B.” Their response addresses only this point: “States 5x − 3 = A(x + 2) + B(x − 1).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: States 5x − 3 = A(x + 2) + B(x − 1). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes x = 1 to find A: 5(1) − 3 = A(3). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Obtains A = 2/3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Substitutes x = −2 to find B: 5(−2) − 3 = B(−3). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Obtains B = 13/3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: States 5x − 3 = A(x + 2) + B(x − 1).
    • Identifies the missing requirement: Substitutes x = 1 to find A: 5(1) − 3 = A(3).
    • Identifies the missing requirement: Obtains A = 2/3.
    • Identifies the missing requirement: Substitutes x = −2 to find B: 5(−2) − 3 = B(−3).
    • Identifies the missing requirement: Obtains B = 13/3.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  19. 19.

    The graph of y = f(x) is transformed to give y = 2f(x − 3) − 1. Describe, in order, the three transformations applied to the graph of y = f(x).

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
    2. Work through this mathematical step: States a horizontal translation of 3 units in the positive x-direction (right), giving f(x − 3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States a vertical stretch with scale factor 2 (relative to the x-axis), giving 2f(x − 3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States a vertical translation of 1 unit downward, giving 2f(x − 3) − 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Transformations inside the function (affecting x) and outside the function (affecting y) can always be identified and described separately, and the order given here matches the order in which they are conventionally applied to the graph.

    Marking points

    • States a horizontal translation of 3 units in the positive x-direction (right), giving f(x − 3).
    • States a vertical stretch with scale factor 2 (relative to the x-axis), giving 2f(x − 3).
    • States a vertical translation of 1 unit downward, giving 2f(x − 3) − 1.

    Examiner tip: Transformations inside the function (affecting x) and outside the function (affecting y) can always be identified and described separately, and the order given here matches the order in which they are conventionally applied to the graph.

  20. 20.

    Marking analysis: A learner attempts the following task: “The graph of y = f(x) is transformed to give y = 2f(x − 3) − 1. Describe, in order, the three transformations applied to the graph of y = f(x).” Their response addresses only this point: “States a horizontal translation of 3 units in the positive x-direction (right), giving f(x − 3).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: States a horizontal translation of 3 units in the positive x-direction (right), giving f(x − 3). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: States a vertical stretch with scale factor 2 (relative to the x-axis), giving 2f(x − 3). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States a vertical translation of 1 unit downward, giving 2f(x − 3) − 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: States a horizontal translation of 3 units in the positive x-direction (right), giving f(x − 3).
    • Identifies the missing requirement: States a vertical stretch with scale factor 2 (relative to the x-axis), giving 2f(x − 3).
    • Identifies the missing requirement: States a vertical translation of 1 unit downward, giving 2f(x − 3) − 1.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  21. 21.

    Solve the equation 3^(2x−1) = 27^(x+2), giving x as an exact value.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Rewrites 27 as 3³, so 27^(x+2) = 3^(3(x+2)). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Equates the exponents, since the bases are equal: 2x − 1 = 3(x + 2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Expands to 2x − 1 = 3x + 6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Solves to obtain x = −7. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Whenever both sides of an exponential equation can be written with the same base, the exponents can simply be equated — always look for a common base before taking logarithms.

    Marking points

    • Rewrites 27 as 3³, so 27^(x+2) = 3^(3(x+2)).
    • Equates the exponents, since the bases are equal: 2x − 1 = 3(x + 2).
    • Expands to 2x − 1 = 3x + 6.
    • Solves to obtain x = −7.

    Examiner tip: Whenever both sides of an exponential equation can be written with the same base, the exponents can simply be equated — always look for a common base before taking logarithms.

  22. 22.

    Marking analysis: A learner attempts the following task: “Solve the equation 3^(2x−1) = 27^(x+2), giving x as an exact value.” Their response addresses only this point: “Rewrites 27 as 3³, so 27^(x+2) = 3^(3(x+2)).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: Rewrites 27 as 3³, so 27^(x+2) = 3^(3(x+2)). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Equates the exponents, since the bases are equal: 2x − 1 = 3(x + 2). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Expands to 2x − 1 = 3x + 6. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Solves to obtain x = −7. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: Rewrites 27 as 3³, so 27^(x+2) = 3^(3(x+2)).
    • Identifies the missing requirement: Equates the exponents, since the bases are equal: 2x − 1 = 3(x + 2).
    • Identifies the missing requirement: Expands to 2x − 1 = 3x + 6.
    • Identifies the missing requirement: Solves to obtain x = −7.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  23. 23.

    The rational function f(x) = (x² + 1)/(x − 2) has an oblique (slant) asymptote. Use polynomial division to find its equation.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Divides x² + 1 by x − 2 using polynomial long division. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Obtains a quotient of x + 2 and a remainder of 5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Writes f(x) = x + 2 + 5/(x − 2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States that as x → ±∞, the remainder term 5/(x − 2) → 0, so the oblique asymptote is y = x + 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: An oblique asymptote arises whenever the numerator's degree is exactly one more than the denominator's — the quotient from polynomial division (ignoring the remainder) always gives its equation directly.

    Marking points

    • Divides x² + 1 by x − 2 using polynomial long division.
    • Obtains a quotient of x + 2 and a remainder of 5.
    • Writes f(x) = x + 2 + 5/(x − 2).
    • States that as x → ±∞, the remainder term 5/(x − 2) → 0, so the oblique asymptote is y = x + 2.

    Examiner tip: An oblique asymptote arises whenever the numerator's degree is exactly one more than the denominator's — the quotient from polynomial division (ignoring the remainder) always gives its equation directly.

  24. 24.

    Marking analysis: A learner attempts the following task: “The rational function f(x) = (x² + 1)/(x − 2) has an oblique (slant) asymptote. Use polynomial division to find its equation.” Their response addresses only this point: “Divides x² + 1 by x − 2 using polynomial long division.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: Divides x² + 1 by x − 2 using polynomial long division. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Obtains a quotient of x + 2 and a remainder of 5. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Writes f(x) = x + 2 + 5/(x − 2). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: States that as x → ±∞, the remainder term 5/(x − 2) → 0, so the oblique asymptote is y = x + 2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: Divides x² + 1 by x − 2 using polynomial long division.
    • Identifies the missing requirement: Obtains a quotient of x + 2 and a remainder of 5.
    • Identifies the missing requirement: Writes f(x) = x + 2 + 5/(x − 2).
    • Identifies the missing requirement: States that as x → ±∞, the remainder term 5/(x − 2) → 0, so the oblique asymptote is y = x + 2.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  25. 25.

    Solve the equation |2x − 5| = 7.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Sets up the two cases: 2x − 5 = 7 and 2x − 5 = −7. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Solves the first case to obtain x = 6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Solves the second case to obtain x = −1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: An absolute value equation |A| = k (for k > 0) always splits into exactly two linear cases, A = k and A = −k — solve both, since both are generally valid solutions.

    Marking points

    • Sets up the two cases: 2x − 5 = 7 and 2x − 5 = −7.
    • Solves the first case to obtain x = 6.
    • Solves the second case to obtain x = −1.

    Examiner tip: An absolute value equation |A| = k (for k > 0) always splits into exactly two linear cases, A = k and A = −k — solve both, since both are generally valid solutions.

  26. 26.

    Marking analysis: A learner attempts the following task: “Solve the equation |2x − 5| = 7.” Their response addresses only this point: “Sets up the two cases: 2x − 5 = 7 and 2x − 5 = −7.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: Sets up the two cases: 2x − 5 = 7 and 2x − 5 = −7. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Solves the first case to obtain x = 6. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Solves the second case to obtain x = −1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: Sets up the two cases: 2x − 5 = 7 and 2x − 5 = −7.
    • Identifies the missing requirement: Solves the first case to obtain x = 6.
    • Identifies the missing requirement: Solves the second case to obtain x = −1.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  27. 27.

    The polynomial f(x) = x³ + ax² − 5x + 6 has (x − 2) as a factor. Find the value of a.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the factor theorem: since (x − 2) is a factor, f(2) = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes x = 2: 8 + 4a − 10 + 6 = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Solves to obtain a = −1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The factor theorem converts a factorisation condition directly into a solvable equation for the unknown coefficient — substitute the root that makes the known factor zero.

    Marking points

    • States the factor theorem: since (x − 2) is a factor, f(2) = 0.
    • Substitutes x = 2: 8 + 4a − 10 + 6 = 0.
    • Solves to obtain a = −1.

    Examiner tip: The factor theorem converts a factorisation condition directly into a solvable equation for the unknown coefficient — substitute the root that makes the known factor zero.

  28. 28.

    Marking analysis: A learner attempts the following task: “The polynomial f(x) = x³ + ax² − 5x + 6 has (x − 2) as a factor. Find the value of a.” Their response addresses only this point: “States the factor theorem: since (x − 2) is a factor, f(2) = 0.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: States the factor theorem: since (x − 2) is a factor, f(2) = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes x = 2: 8 + 4a − 10 + 6 = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Solves to obtain a = −1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: States the factor theorem: since (x − 2) is a factor, f(2) = 0.
    • Identifies the missing requirement: Substitutes x = 2: 8 + 4a − 10 + 6 = 0.
    • Identifies the missing requirement: Solves to obtain a = −1.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

  29. 29.

    Given f(x) = 2x + 1 and g(x) = x² − 3, find (g∘f)(x) and hence solve (g∘f)(x) = 6.

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Forms the composite: (g∘f)(x) = g(2x + 1) = (2x + 1)² − 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Expands to obtain (g∘f)(x) = 4x² + 4x − 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Sets 4x² + 4x − 2 = 6 and simplifies to x² + x − 2 = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Factorises (x + 2)(x − 1) = 0 to obtain x = −2 or x = 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Always fully expand and simplify a composite function before setting it equal to a target value — attempting to solve the unexpanded form directly is far more error-prone.

    Marking points

    • Forms the composite: (g∘f)(x) = g(2x + 1) = (2x + 1)² − 3.
    • Expands to obtain (g∘f)(x) = 4x² + 4x − 2.
    • Sets 4x² + 4x − 2 = 6 and simplifies to x² + x − 2 = 0.
    • Factorises (x + 2)(x − 1) = 0 to obtain x = −2 or x = 1.

    Examiner tip: Always fully expand and simplify a composite function before setting it equal to a target value — attempting to solve the unexpanded form directly is far more error-prone.

  30. 30.

    Marking analysis: A learner attempts the following task: “Given f(x) = 2x + 1 and g(x) = x² − 3, find (g∘f)(x) and hence solve (g∘f)(x) = 6.” Their response addresses only this point: “Forms the composite: (g∘f)(x) = g(2x + 1) = (2x + 1)² − 3.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: Forms the composite: (g∘f)(x) = g(2x + 1) = (2x + 1)² − 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Expands to obtain (g∘f)(x) = 4x² + 4x − 2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Sets 4x² + 4x − 2 = 6 and simplifies to x² + x − 2 = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Factorises (x + 2)(x − 1) = 0 to obtain x = −2 or x = 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: Forms the composite: (g∘f)(x) = g(2x + 1) = (2x + 1)² − 3.
    • Identifies the missing requirement: Expands to obtain (g∘f)(x) = 4x² + 4x − 2.
    • Identifies the missing requirement: Sets 4x² + 4x − 2 = 6 and simplifies to x² + x − 2 = 0.
    • Identifies the missing requirement: Factorises (x + 2)(x − 1) = 0 to obtain x = −2 or x = 1.

    Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.