School / IB / MATH AA HL / Functions Question practice
Functions About this practice HL-only depth: polynomial division, the factor theorem and rational functions.
Mathematics AA: Higher Level Functions
View All Incomplete Complete Review mistakes One at a time
Revision Ladder All stages Easy Medium HardSoon
Paper All papers Paper 1 style Paper 2 style General practice (not paper-specific)
About paper groupings Paper labels are an unofficial, independently authored grouping, applied only where a question's own format genuinely matches a real paper convention (such as IB Mathematics AA's non-calculator/calculator split). They do not reproduce any exam board's real paper numbering or mark allocation, and uncertain questions are labelled general practice instead.
1 Show that (x − 2) is a factor of p(x) = x³ − 3x² − 4x + 12, and hence fully factorise p(x). Paper 1 style Medium 5 marks No calculator + 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. Marking analysis Medium 5 marks No calculator + 3 The rational function f(x) = (2x + 1)/(x − 3) has a vertical and a horizontal asymptote. State the equation of each asymptote. Paper 1 style Easy 3 marks No calculator + 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. Marking analysis Easy 3 marks No calculator + 5 The cubic equation 2x³ − 5x² + 3x − 1 = 0 has roots α, β, γ. Find α + β + γ and αβγ. Paper 1 style Medium 4 marks No calculator + 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. Marking analysis Medium 4 marks No calculator + 7 The function f(x) = √(x + 4) has domain x ≥ −4. Find f⁻¹(x) and state its domain and range. Paper 1 style Medium 5 marks No calculator + 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. Marking analysis Medium 5 marks No calculator + 9 Solve the inequality (x − 1)(x + 2)(x − 3) ≥ 0 using a sign diagram. Paper 1 style Medium 5 marks No calculator + 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. Marking analysis Medium 5 marks No calculator + 11 Show algebraically that f(x) = (x + 1)/(x − 1), x ≠ 1, is a self-inverse function. Paper 1 style Medium 5 marks No calculator + 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. Marking analysis Medium 5 marks No calculator + 13 Given f(x) = √x and g(x) = x − 4, find the domain of the composite function (f ∘ g)(x). Paper 1 style Medium 4 marks No calculator + 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. Marking analysis Medium 4 marks No calculator + 15 Determine algebraically whether f(x) = x³ − 2x is odd, even, or neither. Paper 1 style Medium 4 marks No calculator + 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. Marking analysis Medium 4 marks No calculator + 17 Express (5x − 3)/((x − 1)(x + 2)) in the form A/(x − 1) + B/(x + 2), finding the values of A and B. Paper 1 style Medium 5 marks No calculator + 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. Marking analysis Medium 5 marks No calculator + 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). Paper 1 style Easy 3 marks No calculator + 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. Marking analysis Easy 3 marks No calculator + 21 Solve the equation 3^(2x−1) = 27^(x+2), giving x as an exact value. Solve the equation 3^(2x−1) = 27(x+2) , giving x as an exact value. Paper 1 style Medium 4 marks No calculator + 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. 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. Marking analysis Medium 4 marks No calculator + 23 The rational function f(x) = (x² + 1)/(x − 2) has an oblique (slant) asymptote. Use polynomial division to find its equation. Paper 1 style Medium 4 marks No calculator + 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. Marking analysis Medium 4 marks No calculator + 25 Solve the equation |2x − 5| = 7. Paper 1 style Easy 3 marks No calculator + 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. Marking analysis Easy 3 marks No calculator + 27 The polynomial f(x) = x³ + ax² − 5x + 6 has (x − 2) as a factor. Find the value of a. Paper 2 style Easy 3 marks Calculator + 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. Marking analysis Easy 3 marks Calculator + 29 Given f(x) = 2x + 1 and g(x) = x² − 3, find (g∘f)(x) and hence solve (g∘f)(x) = 6. Paper 2 style Medium 4 marks Calculator + 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. Marking analysis Medium 4 marks Calculator + Self-assessed Not marked yet
Set total 122
This is a self-study tool, not a certified answer key or a predicted IB grade.
Loading progress Reset progress
Before you practise
Questions learners ask about this practice Are these official IB questions? No. These are original SubjectScout practice questions for Functions. They are not official past-paper questions or endorsed material.
What does this Mathematics AA: Higher Level practice page include? It includes selected topic questions, marking points, and feedback prompts designed to help learners practise before requesting teacher support.
Can I request a teacher for this exact topic? Yes. Tell SubjectScout the subject, exact topic, level and deadline, and the team will try to match you with a suitable verified teacher.
Unofficial content under accuracy, provenance, and rights review. Not affiliated with or endorsed by the International Baccalaureate Organization. Official past-paper questions are not reproduced here.
