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IB · MATH AA SL

Mathematics AA: Standard Level

Functions — Topic 2

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

    The function f(x) = x² − 4x + 7 is defined for all real x. Write f(x) in completed-square form. Hence state the coordinates of the vertex and the minimum value of f.

    [3 marks] · no calculator

    Answer explanation

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

    1. Start by grouping the x-terms: x² − 4x + 7 = (x² − 4x) + 7.
    2. Complete the square: x² − 4x = (x − 2)² − 4, so f(x) = (x − 2)² − 4 + 7.
    3. Simplify to f(x) = (x − 2)² + 3. Since the squared term is never negative, the minimum occurs when x = 2.
    4. At x = 2, f(x) = 3, so the vertex is (2, 3) and the minimum value is 3.

    Marking points

    • Correctly obtains f(x) = (x − 2)² + 3.
    • States the vertex as (2, 3).
    • States the minimum value as 3.

    Examiner tip: Use the completed-square form to read both answers directly; do not differentiate unless asked.

  2. 2.

    Marking analysis: A learner attempts the following task: “The function f(x) = x² − 4x + 7 is defined for all real x. Write f(x) in completed-square form. Hence state the coordinates of the vertex and the minimum value of f.” Their response addresses only this point: “Correctly obtains f(x) = (x − 2)² + 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: Correctly obtains f(x) = (x − 2)² + 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: States the vertex as (2, 3). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States the minimum value as 3. 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: Correctly obtains f(x) = (x − 2)² + 3.
    • Identifies the missing requirement: States the vertex as (2, 3).
    • Identifies the missing requirement: States the minimum value as 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.

  3. 3.

    A population is modelled by P(t) = 1200(1.08)^t, where t is the number of complete years after the start of 2026. Find the first value of t for which the population exceeds 2000. Show your method.

    [4 marks]

    Answer explanation

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

    1. The population first exceeds 2000 when 1200(1.08)^t > 2000.
    2. Divide both sides by 1200 to get (1.08)^t > 2000/1200.
    3. Take natural logarithms: t ln(1.08) > ln(2000/1200), so t > ln(2000/1200) / ln(1.08).
    4. This gives t > 6.64 approximately. Because t counts complete years, the first whole value is t = 7.

    Marking points

    • Sets up 1200(1.08)^t > 2000 or the corresponding equality.
    • Uses logarithms: t > ln(2000/1200) ÷ ln(1.08).
    • Obtains t > 6.64 approximately.
    • States the first complete-year value t = 7.

    Examiner tip: The question asks for complete years, so the decimal solution must be rounded up.

  4. 4.

    Marking analysis: A learner attempts the following task: “A population is modelled by P(t) = 1200(1.08)^t, where t is the number of complete years after the start of 2026. Find the first value of t for which the population exceeds 2000. Show your method.” Their response addresses only this point: “Sets up 1200(1.08)^t > 2000 or the corresponding equality.” 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: Sets up 1200(1.08)^t > 2000 or the corresponding equality. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Uses logarithms: t > ln(2000/1200) ÷ ln(1.08). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Obtains t > 6.64 approximately. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: States the first complete-year value t = 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: Sets up 1200(1.08)^t > 2000 or the corresponding equality.
    • Identifies the missing requirement: Uses logarithms: t > ln(2000/1200) ÷ ln(1.08).
    • Identifies the missing requirement: Obtains t > 6.64 approximately.
    • Identifies the missing requirement: States the first complete-year value t = 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.

  5. 5.

    Solve the inequality x² − 5x + 6 ≤ 0. Give your answer using interval notation.

    [3 marks] · no calculator

    Answer explanation

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

    1. Factorise the quadratic: x² − 5x + 6 = (x − 2)(x − 3).
    2. The critical values are x = 2 and x = 3, where the expression equals zero.
    3. Because the coefficient of x² is positive, the parabola opens upwards, so the expression is below or equal to zero between the roots.
    4. Include the endpoints because the inequality is ≤ 0, giving [2, 3].

    Marking points

    • Factorises to (x − 2)(x − 3) ≤ 0.
    • Identifies critical values x = 2 and x = 3.
    • States the solution in interval notation: [2, 3] (equivalently 2 ≤ x ≤ 3).

    Examiner tip: A quadratic with positive leading coefficient and two distinct real roots is negative between the roots and zero at them. Include both endpoints for ≤ 0.

  6. 6.

    Marking analysis: A learner attempts the following task: “Solve the inequality x² − 5x + 6 ≤ 0. Give your answer using interval notation.” Their response addresses only this point: “Factorises to (x − 2)(x − 3) ≤ 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] · 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: Factorises to (x − 2)(x − 3) ≤ 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Identifies critical values x = 2 and x = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States the solution in interval notation: [2, 3] (equivalently 2 ≤ x ≤ 3). 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: Factorises to (x − 2)(x − 3) ≤ 0.
    • Identifies the missing requirement: Identifies critical values x = 2 and x = 3.
    • Identifies the missing requirement: States the solution in interval notation: [2, 3] (equivalently 2 ≤ 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.

  7. 7.

    The function f(x) = 3x − 2 is defined for all real x. Find f⁻¹(x) and state its domain.

    [3 marks] · no calculator

    Answer explanation

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

    1. An inverse reverses the input-output relationship. Write y = 3x - 2, then exchange input and output: x = 3y - 2.
    2. Add 2 to both sides and divide by 3: y = (x + 2)/3. This is f⁻¹(x).
    3. The original non-horizontal straight line has every real value in its range, so the inverse accepts every real input. Check f((x + 2)/3) = x.

    Marking points

    • Sets y = 3x − 2 and swaps x and y (or equivalent method).
    • Rearranges to obtain f⁻¹(x) = (x + 2)/3.
    • States the domain of f⁻¹ as all real numbers.

    Examiner tip: Here the linear function has non-zero gradient and domain ℝ, so its range and the domain of its inverse are both ℝ.

  8. 8.

    Marking analysis: A learner attempts the following task: “The function f(x) = 3x − 2 is defined for all real x. Find f⁻¹(x) and state its domain.” Their response addresses only this point: “Sets y = 3x − 2 and swaps x and y (or equivalent method).” 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 y = 3x − 2 and swaps x and y (or equivalent method). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Rearranges to obtain f⁻¹(x) = (x + 2)/3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States the domain of f⁻¹ as all real numbers. 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 y = 3x − 2 and swaps x and y (or equivalent method).
    • Identifies the missing requirement: Rearranges to obtain f⁻¹(x) = (x + 2)/3.
    • Identifies the missing requirement: States the domain of f⁻¹ as all real numbers.

    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.

    Let f(x) = 2x + 1 and g(x) = x². Find an expression for the composite function (f ∘ g)(x) and evaluate (f ∘ g)(3).

    [3 marks] · no calculator

    Answer explanation

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

    1. Build a supported judgement: identify the claim, use the question's evidence, consider a relevant limitation or alternative, and make the conclusion depend on that evidence. There may be more than one defensible answer.
    2. Work through this mathematical step: Substitutes g(x) into f to obtain f(g(x)) = 2x² + 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 = 3 into the composite function. 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 (f ∘ g)(3) = 19. 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: (f ∘ g)(x) means apply g first, then f — substitute g(x) into f, not the reverse order.

    Marking points

    • Substitutes g(x) into f to obtain f(g(x)) = 2x² + 1.
    • Substitutes x = 3 into the composite function.
    • Obtains (f ∘ g)(3) = 19.

    Examiner tip: (f ∘ g)(x) means apply g first, then f — substitute g(x) into f, not the reverse order.

  10. 10.

    Marking analysis: A learner attempts the following task: “Let f(x) = 2x + 1 and g(x) = x². Find an expression for the composite function (f ∘ g)(x) and evaluate (f ∘ g)(3).” Their response addresses only this point: “Substitutes g(x) into f to obtain f(g(x)) = 2x² + 1.” 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: Substitutes g(x) into f to obtain f(g(x)) = 2x² + 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes x = 3 into the composite function. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Obtains (f ∘ g)(3) = 19. 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: Substitutes g(x) into f to obtain f(g(x)) = 2x² + 1.
    • Identifies the missing requirement: Substitutes x = 3 into the composite function.
    • Identifies the missing requirement: Obtains (f ∘ g)(3) = 19.

    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.

    Solve the equation log₂(x) + log₂(x − 2) = 3 for x, stating why one algebraic solution must be rejected.

    [5 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: Uses the log law log₂(x) + log₂(x − 2) = log₂[x(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: Sets x(x − 2) = 2³ = 8. 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: Forms x² − 2x − 8 = 0 and factorises to (x − 4)(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: Obtains candidate solutions x = 4 and 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: The real-domain conditions x > 0 and x − 2 > 0 require x > 2. Rejects x = −2 and states x = 4. 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: Always check log-equation solutions against the original domain restrictions before finalising an answer.

    Marking points

    • Uses the log law log₂(x) + log₂(x − 2) = log₂[x(x − 2)].
    • Sets x(x − 2) = 2³ = 8.
    • Forms x² − 2x − 8 = 0 and factorises to (x − 4)(x + 2) = 0.
    • Obtains candidate solutions x = 4 and x = −2.
    • The real-domain conditions x > 0 and x − 2 > 0 require x > 2. Rejects x = −2 and states x = 4.

    Examiner tip: Always check log-equation solutions against the original domain restrictions before finalising an answer.

  12. 12.

    Marking analysis: A learner attempts the following task: “Solve the equation log₂(x) + log₂(x − 2) = 3 for x, stating why one algebraic solution must be rejected.” Their response addresses only this point: “Uses the log law log₂(x) + log₂(x − 2) = log₂[x(x − 2)].” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [5 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: Uses the log law log₂(x) + log₂(x − 2) = log₂[x(x − 2)]. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Sets x(x − 2) = 2³ = 8. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Forms x² − 2x − 8 = 0 and factorises to (x − 4)(x + 2) = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains candidate solutions x = 4 and x = −2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: The real-domain conditions x > 0 and x − 2 > 0 require x > 2. Rejects x = −2 and states x = 4. 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: Uses the log law log₂(x) + log₂(x − 2) = log₂[x(x − 2)].
    • Identifies the missing requirement: Sets x(x − 2) = 2³ = 8.
    • Identifies the missing requirement: Forms x² − 2x − 8 = 0 and factorises to (x − 4)(x + 2) = 0.
    • Identifies the missing requirement: Obtains candidate solutions x = 4 and x = −2.
    • Identifies the missing requirement: The real-domain conditions x > 0 and x − 2 > 0 require x > 2. Rejects x = −2 and states 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.

  13. 13.

    A bacteria culture grows according to N(t) = 500(2)^(t/3), where t is measured in hours. Find the initial population and calculate how long it takes for the population to reach 4000.

    [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: States the initial population N(0) = 500. 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: Sets up 500(2)^(t/3) = 4000. 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: Simplifies to 2^(t/3) = 8 and uses t/3 = 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: Obtains t = 9 hours. 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: Recognising 8 as a power of 2 avoids needing logarithms for this particular equation.

    Marking points

    • States the initial population N(0) = 500.
    • Sets up 500(2)^(t/3) = 4000.
    • Simplifies to 2^(t/3) = 8 and uses t/3 = 3.
    • Obtains t = 9 hours.

    Examiner tip: Recognising 8 as a power of 2 avoids needing logarithms for this particular equation.

  14. 14.

    Marking analysis: A learner attempts the following task: “A bacteria culture grows according to N(t) = 500(2)^(t/3), where t is measured in hours. Find the initial population and calculate how long it takes for the population to reach 4000.” Their response addresses only this point: “States the initial population N(0) = 500.” 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: States the initial population N(0) = 500. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Sets up 500(2)^(t/3) = 4000. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Simplifies to 2^(t/3) = 8 and uses t/3 = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains t = 9 hours. 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 initial population N(0) = 500.
    • Identifies the missing requirement: Sets up 500(2)^(t/3) = 4000.
    • Identifies the missing requirement: Simplifies to 2^(t/3) = 8 and uses t/3 = 3.
    • Identifies the missing requirement: Obtains t = 9 hours.

    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.

    The graph of y = f(x) is transformed to give the graph of y = f(x − 2) + 3. Describe fully the geometric transformation applied to the graph of f.

    [2 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 a horizontal translation of 2 units in the positive x-direction. 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 vertical translation of 3 units in the positive y-direction. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: f(x − a) shifts right by a; +b outside the function shifts up by b — the signs are easy to reverse by mistake.

    Marking points

    • Identifies a horizontal translation of 2 units in the positive x-direction.
    • Identifies a vertical translation of 3 units in the positive y-direction.

    Examiner tip: f(x − a) shifts right by a; +b outside the function shifts up by b — the signs are easy to reverse by mistake.

  16. 16.

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

    [2 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 a horizontal translation of 2 units in the positive x-direction. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Identifies a vertical translation of 3 units in the positive y-direction. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. 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 a horizontal translation of 2 units in the positive x-direction.
    • Identifies the missing requirement: Identifies a vertical translation of 3 units in the positive y-direction.

    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.

    Solve the equation e^(2x) = 15, giving your answer to three significant figures.

    [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: Takes the natural logarithm of both sides: 2x = ln(15). 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: Rearranges to the exact expression x = ln(15)/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: Obtains x ≈ 1.35. 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: Taking ln of both sides is the standard method whenever the unknown is in the exponent of e.

    Marking points

    • Takes the natural logarithm of both sides: 2x = ln(15).
    • Rearranges to the exact expression x = ln(15)/2.
    • Obtains x ≈ 1.35.

    Examiner tip: Taking ln of both sides is the standard method whenever the unknown is in the exponent of e.

  18. 18.

    Marking analysis: A learner attempts the following task: “Solve the equation e^(2x) = 15, giving your answer to three significant figures.” Their response addresses only this point: “Takes the natural logarithm of both sides: 2x = ln(15).” 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: Takes the natural logarithm of both sides: 2x = ln(15). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Rearranges to the exact expression x = ln(15)/2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Obtains x ≈ 1.35. 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: Takes the natural logarithm of both sides: 2x = ln(15).
    • Identifies the missing requirement: Rearranges to the exact expression x = ln(15)/2.
    • Identifies the missing requirement: Obtains x ≈ 1.35.

    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.

    Find the largest possible domain of f(x) = √(x − 3), and state the corresponding range.

    [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: Requires the expression under the root to be non-negative: x − 3 ≥ 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: States the domain as 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 the range as f(x) ≥ 0. 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: A square root function's output is never negative, which directly gives the range once the domain is found.

    Marking points

    • Requires the expression under the root to be non-negative: x − 3 ≥ 0.
    • States the domain as x ≥ 3.
    • States the range as f(x) ≥ 0.

    Examiner tip: A square root function's output is never negative, which directly gives the range once the domain is found.

  20. 20.

    Marking analysis: A learner attempts the following task: “Find the largest possible domain of f(x) = √(x − 3), and state the corresponding range.” Their response addresses only this point: “Requires the expression under the root to be non-negative: x − 3 ≥ 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] · 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: Requires the expression under the root to be non-negative: x − 3 ≥ 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: States the domain as x ≥ 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States the range as f(x) ≥ 0. 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: Requires the expression under the root to be non-negative: x − 3 ≥ 0.
    • Identifies the missing requirement: States the domain as x ≥ 3.
    • Identifies the missing requirement: States the range as f(x) ≥ 0.

    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.

    The quadratic equation kx² + 4x + 1 = 0 has two equal real roots. Find the value of k.

    [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 that for equal roots, the discriminant b² − 4ac = 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 4² − 4(k)(1) = 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 k = 4. 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: A zero discriminant is the exact condition for a quadratic to have a repeated (double) root — the parabola touches the x-axis at exactly one point instead of crossing it.

    Marking points

    • States that for equal roots, the discriminant b² − 4ac = 0.
    • Substitutes 4² − 4(k)(1) = 0.
    • Solves to obtain k = 4.

    Examiner tip: A zero discriminant is the exact condition for a quadratic to have a repeated (double) root — the parabola touches the x-axis at exactly one point instead of crossing it.

  22. 22.

    Marking analysis: A learner attempts the following task: “The quadratic equation kx² + 4x + 1 = 0 has two equal real roots. Find the value of k.” Their response addresses only this point: “States that for equal roots, the discriminant b² − 4ac = 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 that for equal roots, the discriminant b² − 4ac = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes 4² − 4(k)(1) = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Solves to obtain k = 4. 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 that for equal roots, the discriminant b² − 4ac = 0.
    • Identifies the missing requirement: Substitutes 4² − 4(k)(1) = 0.
    • Identifies the missing requirement: Solves to obtain k = 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.

  23. 23.

    A parabola has vertex (1, −4) and passes through the point (3, 4). Find the equation of the parabola in the form y = a(x − h)² + k.

    [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: Writes the vertex form using the given vertex: y = a(x − 1)² − 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: Substitutes the point (3, 4): 4 = a(3 − 1)² − 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: Simplifies to 8 = 4a and solves to obtain a = 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 the equation y = 2(x − 1)² − 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: Vertex form y = a(x − h)² + k is the fastest route when the vertex is already known — substitute the vertex first, then use any other given point to solve for the single remaining unknown, a.

    Marking points

    • Writes the vertex form using the given vertex: y = a(x − 1)² − 4.
    • Substitutes the point (3, 4): 4 = a(3 − 1)² − 4.
    • Simplifies to 8 = 4a and solves to obtain a = 2.
    • States the equation y = 2(x − 1)² − 4.

    Examiner tip: Vertex form y = a(x − h)² + k is the fastest route when the vertex is already known — substitute the vertex first, then use any other given point to solve for the single remaining unknown, a.

  24. 24.

    Marking analysis: A learner attempts the following task: “A parabola has vertex (1, −4) and passes through the point (3, 4). Find the equation of the parabola in the form y = a(x − h)² + k.” Their response addresses only this point: “Writes the vertex form using the given vertex: y = a(x − 1)² − 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]

    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 the vertex form using the given vertex: y = a(x − 1)² − 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes the point (3, 4): 4 = a(3 − 1)² − 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Simplifies to 8 = 4a and solves to obtain a = 2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: States the equation y = 2(x − 1)² − 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: Writes the vertex form using the given vertex: y = a(x − 1)² − 4.
    • Identifies the missing requirement: Substitutes the point (3, 4): 4 = a(3 − 1)² − 4.
    • Identifies the missing requirement: Simplifies to 8 = 4a and solves to obtain a = 2.
    • Identifies the missing requirement: States the equation y = 2(x − 1)² − 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.

  25. 25.

    A radioactive sample decays according to the model M(t) = 80e^(−0.05t), where M is the mass in grams and t is the time in years. Find the initial mass, and calculate the time taken for the mass to decay to 20 grams.

    [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: States the initial mass M(0) = 80 g. 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: Sets up 80e^(−0.05t) = 20. 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: Simplifies to e^(−0.05t) = 0.25 and takes the natural logarithm: −0.05t = ln(0.25). 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 t ≈ 27.7 years. 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: A negative exponent (or negative rate constant) always indicates decay rather than growth — the model still uses exactly the same logarithm technique to solve for time.

    Marking points

    • States the initial mass M(0) = 80 g.
    • Sets up 80e^(−0.05t) = 20.
    • Simplifies to e^(−0.05t) = 0.25 and takes the natural logarithm: −0.05t = ln(0.25).
    • Solves to obtain t ≈ 27.7 years.

    Examiner tip: A negative exponent (or negative rate constant) always indicates decay rather than growth — the model still uses exactly the same logarithm technique to solve for time.

  26. 26.

    Marking analysis: A learner attempts the following task: “A radioactive sample decays according to the model M(t) = 80e^(−0.05t), where M is the mass in grams and t is the time in years. Find the initial mass, and calculate the time taken for the mass to decay to 20 grams.” Their response addresses only this point: “States the initial mass M(0) = 80 g.” 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: States the initial mass M(0) = 80 g. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Sets up 80e^(−0.05t) = 20. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Simplifies to e^(−0.05t) = 0.25 and takes the natural logarithm: −0.05t = ln(0.25). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Solves to obtain t ≈ 27.7 years. 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 initial mass M(0) = 80 g.
    • Identifies the missing requirement: Sets up 80e^(−0.05t) = 20.
    • Identifies the missing requirement: Simplifies to e^(−0.05t) = 0.25 and takes the natural logarithm: −0.05t = ln(0.25).
    • Identifies the missing requirement: Solves to obtain t ≈ 27.7 years.

    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.

    Solve the equation 2ln(x) − ln(x − 1) = ln(4) for x, given that x > 1.

    [5 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: Uses the power law to rewrite 2ln(x) as ln(x²). 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: Combines the left side using the quotient law: ln(x²/(x − 1)) = ln(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: Equates the arguments (since ln is one-to-one): x²/(x − 1) = 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: Forms and factorises x² − 4x + 4 = 0 as (x − 2)² = 0. 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 x = 2, and confirms this satisfies the domain condition x > 1. 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: Combine all logarithm terms into a single logarithm first using the log laws, then equate the arguments directly — this is far more reliable than trying to isolate x while logarithms remain separated.

    Marking points

    • Uses the power law to rewrite 2ln(x) as ln(x²).
    • Combines the left side using the quotient law: ln(x²/(x − 1)) = ln(4).
    • Equates the arguments (since ln is one-to-one): x²/(x − 1) = 4.
    • Forms and factorises x² − 4x + 4 = 0 as (x − 2)² = 0.
    • Obtains x = 2, and confirms this satisfies the domain condition x > 1.

    Examiner tip: Combine all logarithm terms into a single logarithm first using the log laws, then equate the arguments directly — this is far more reliable than trying to isolate x while logarithms remain separated.

  28. 28.

    Marking analysis: A learner attempts the following task: “Solve the equation 2ln(x) − ln(x − 1) = ln(4) for x, given that x > 1.” Their response addresses only this point: “Uses the power law to rewrite 2ln(x) as ln(x²).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [5 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: Uses the power law to rewrite 2ln(x) as ln(x²). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Combines the left side using the quotient law: ln(x²/(x − 1)) = ln(4). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Equates the arguments (since ln is one-to-one): x²/(x − 1) = 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Forms and factorises x² − 4x + 4 = 0 as (x − 2)² = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Obtains x = 2, and confirms this satisfies the domain condition x > 1. 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: Uses the power law to rewrite 2ln(x) as ln(x²).
    • Identifies the missing requirement: Combines the left side using the quotient law: ln(x²/(x − 1)) = ln(4).
    • Identifies the missing requirement: Equates the arguments (since ln is one-to-one): x²/(x − 1) = 4.
    • Identifies the missing requirement: Forms and factorises x² − 4x + 4 = 0 as (x − 2)² = 0.
    • Identifies the missing requirement: Obtains x = 2, and confirms this satisfies the domain condition 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.

  29. 29.

    State the domain of the function f(x) = 1/(x − 3), and write the equations of its vertical and horizontal asymptotes.

    [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 the domain as all real x except x = 3 (x ≠ 3), since division by zero is undefined. 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 the vertical asymptote as 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 the horizontal asymptote as y = 0, since f(x) → 0 as x → ±∞. 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 simple reciprocal-type function, the vertical asymptote always occurs exactly where the denominator equals zero, and the excluded domain value is always the same as the vertical asymptote's x-coordinate.

    Marking points

    • States the domain as all real x except x = 3 (x ≠ 3), since division by zero is undefined.
    • States the vertical asymptote as x = 3.
    • States the horizontal asymptote as y = 0, since f(x) → 0 as x → ±∞.

    Examiner tip: For a simple reciprocal-type function, the vertical asymptote always occurs exactly where the denominator equals zero, and the excluded domain value is always the same as the vertical asymptote's x-coordinate.

  30. 30.

    Marking analysis: A learner attempts the following task: “State the domain of the function f(x) = 1/(x − 3), and write the equations of its vertical and horizontal asymptotes.” Their response addresses only this point: “States the domain as all real x except x = 3 (x ≠ 3), since division by zero is undefined.” 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 the domain as all real x except x = 3 (x ≠ 3), since division by zero is undefined. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: States the vertical asymptote as x = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States the horizontal asymptote as y = 0, since f(x) → 0 as x → ±∞. 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 domain as all real x except x = 3 (x ≠ 3), since division by zero is undefined.
    • Identifies the missing requirement: States the vertical asymptote as x = 3.
    • Identifies the missing requirement: States the horizontal asymptote as y = 0, since f(x) → 0 as x → ±∞.

    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.

  31. 31.

    f(x) = 3x - 5 for x >= 2. Find its inverse and state the domain of the inverse.

    [3 marks] · no calculator

    Answer explanation

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

    1. The function is increasing, so its lowest allowed input maps to the lowest output, f(2) = 1.
    2. Solve for the input, then exchange variable names. An inverse can accept exactly the outputs the original function produces.

    Marking points

    • The range of f is y >= 1.
    • Rearranging y = 3x - 5 gives x = (y + 5)/3.
    • f^-1(x) = (x + 5)/3, with domain x >= 1.

    Examiner tip: The inverse domain comes from the original range, not its original domain.

  32. 32.

    Marking analysis: A learner attempts the following task: “f(x) = 3x - 5 for x >= 2. Find its inverse and state the domain of the inverse.” Their response addresses only this point: “The range of f is y >= 1.” 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: The range of f is y >= 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Rearranging y = 3x - 5 gives x = (y + 5)/3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: f^-1(x) = (x + 5)/3, with domain 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: The range of f is y >= 1.
    • Identifies the missing requirement: Rearranging y = 3x - 5 gives x = (y + 5)/3.
    • Identifies the missing requirement: f^-1(x) = (x + 5)/3, with domain 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.