School / IB / MATH AA SL / Functions Question practice
Functions About this practice Quadratics, transformations and exponential models.
Topic overview Quick answer
What this functions practice helps with Use this IB Mathematics AA SL functions set for quadratic functions, composite and inverse functions, logarithms, exponential growth models, and calculator/non-calculator method marks.
Exponential model example: solve 1200(1.08)^t > 2000 by using logarithms, then round up because t is a complete number of years. Paper 1 style: practise completing the square, inequalities, inverse functions, composite functions, and exact algebra without a calculator. Paper 2 style: practise logarithmic and exponential modelling questions where the calculator supports the final numerical decision. Mathematics AA: Standard 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 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. Paper 1 style Easy 3 marks No calculator + 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. Marking analysis Easy 3 marks No calculator + 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. 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. Paper 2 style Medium 4 marks Calculator + 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. 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. Marking analysis Medium 4 marks Calculator + 5 Solve the inequality x² − 5x + 6 ≤ 0. Give your answer using interval notation. Paper 1 style Easy 3 marks No calculator + 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. Marking analysis Easy 3 marks No calculator + 7 The function f(x) = 3x − 2 is defined for all real x. Find f⁻¹(x) and state its domain. Paper 1 style Easy 3 marks No calculator + 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. Marking analysis Easy 3 marks No calculator + 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). Paper 1 style Easy 3 marks No calculator + 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. Marking analysis Easy 3 marks No calculator + 11 Solve the equation log₂(x) + log₂(x − 2) = 3 for x, stating why one algebraic solution must be rejected. Paper 2 style Medium 5 marks Calculator + 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. Marking analysis Medium 5 marks Calculator + 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. Paper 2 style Medium 4 marks Calculator + 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. Marking analysis Medium 4 marks Calculator + 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. Paper 1 style Easy 2 marks No calculator + 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. Marking analysis Easy 2 marks No calculator + 17 Solve the equation e^(2x) = 15, giving your answer to three significant figures. Solve the equation e(2x) = 15, giving your answer to three significant figures. Paper 2 style Easy 3 marks Calculator + 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. 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. Marking analysis Easy 3 marks Calculator + 19 Find the largest possible domain of f(x) = √(x − 3), and state the corresponding range. Paper 1 style Easy 3 marks No calculator + 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. Marking analysis Easy 3 marks No calculator + 21 The quadratic equation kx² + 4x + 1 = 0 has two equal real roots. Find the value of k. Paper 2 style Easy 3 marks Calculator + 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. Marking analysis Easy 3 marks Calculator + 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. Paper 2 style Medium 4 marks Calculator + 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. Marking analysis Medium 4 marks Calculator + 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. Paper 2 style Medium 4 marks Calculator + 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. Marking analysis Medium 4 marks Calculator + 27 Solve the equation 2ln(x) − ln(x − 1) = ln(4) for x, given that x > 1. Paper 2 style Medium 5 marks Calculator + 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. Marking analysis Medium 5 marks Calculator + 29 State the domain of the function f(x) = 1/(x − 3), and write the equations of its vertical and horizontal asymptotes. Paper 1 style Easy 3 marks No calculator + 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. Marking analysis Easy 3 marks No calculator + 31 f(x) = 3x - 5 for x >= 2. Find its inverse and state the domain of the inverse. Paper 1 style Easy 3 marks No calculator + 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. Marking analysis Easy 3 marks No calculator + Self-assessed Not marked yet
Set total 110
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: Standard 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.
