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IB Maths AI HL · Unit 2: Functions

IB Maths AI HL Composite, Inverse and Transformations Questions

Updated October 2026

Exam-style IB Maths AI HL composite, inverse and transformations questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

50 questions
Paper 1: 43
Paper 2: 7
17 easy
17 medium
13 hard
3 very hard
3 worked examples

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What's examined in AI HL composite, inverse and transformations

The question bank covers these composite, inverse and transformations question types (number of questions in brackets):

Function Transformations (39)
Inverse Functions (9)
Composite Functions (2)

Composite, Inverse and Transformations worked examples

Worked example 1: Evaluating Composite Functions · easy

Let $f (x) = 2 x - 5$ and $g (x) = x^{2} + 1$ . Calculate the exact value of $(g \circ f) (3)$ .

Solution

5 steps · 1 to fill in

Examiner tip: A very common mistake is confusing

(g \circ f) (x)

with the product

g (x) \times f (x)

. Always work from the inside bracket outwards.

Worked example 2: Finding the Inverse of a Rational Function · medium

Let $h (x) = \frac{3 x - 5}{2 x + 1}$ for $x \neq - 0.5$ . Find an algebraic expression for the inverse function $h^{- 1} (x)$ .

Solution

6 steps

Examiner tip: When finding the inverse of a rational function, collecting all the

y

terms on one side so you can factorise

y

out is the crucial algebraic step that students frequently miss.

Worked example 3: Constructing Transformed Functions · hard

The graph of $y = \ln (x)$ undergoes a sequence of three geometric transformations: it is translated horizontally by $3$ units to the right, stretched vertically by a scale factor of $2$ , and finally reflected in the $x$ -axis to form the new function $g (x)$ . Find the expression for $g (x)$ and state its exact domain.

Solution

5 steps

Examiner tip: Order matters when applying transformations. Always build your transformed function step-by-step in the exact sequence given in the prompt to avoid sign and scaling errors.

Play with it Translate a parabola: drag the sliders

The dashed curve is y = f(x), where f(x) = x². Change h and k so that y = f(x − h) + k passes through all the points. Write the translation as a vector.

Want more? Open the full Grapher.

Try these IB Maths AI HL composite, inverse and transformations questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 4 marks · Paper 1

Let $f (x) = 2 x - 5$ and $g (x) = x^{2} + 1$ .
(a) Find $(g \circ f) (x)$ . [2 marks]
(b) Calculate $(f \circ g) (3)$ . [2 marks]

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

Find the inverse of the rational function $f (x) = \frac{2 x - 3}{x + 1}$ , where $x \neq - 1$ .

Full worked solution Free, no sign-in

Let $x = \frac{2 y - 3}{y + 1}$ . [M1]
$x (y + 1) = 2 y - 3 ⟹ x y + x = 2 y - 3$ . [M1]
$x y - 2 y = - x - 3 ⟹ y (x - 2) = - x - 3$ . [M1]
$f^{- 1} (x) = \frac{- x - 3}{x - 2}$ or $\frac{x + 3}{2 - x}$ . [A1][A1]

Every question here has a full worked solution. See them all →

Question 3 · hard · 7 marks · Paper 1

Show algebraically that the function $f (x) = \frac{3 x + 2}{x - 3}$ , for $x \neq 3$ , is a self-inverse function.

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FAQ

How many IB Maths AI HL composite, inverse and transformations questions are there?

There are 50 exam-style composite, inverse and transformations questions in the AI HL question bank (Paper 1: 43 · Paper 2: 7), graded 17 easy, 17 medium, 13 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is composite, inverse and transformations on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 43 · Paper 2: 7. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI HL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

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Free ready-to-teach lessons (slides and teacher notes): Vertical + horizontal shifts · Domain, range, composite & inverse functions.

IB Maths AI HL Composite, Inverse and Transformations Questions

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What's examined in AI HL composite, inverse and transformations

Composite, Inverse and Transformations worked examples

Worked example 1: Evaluating Composite Functions · easy

Solution

Worked example 2: Finding the Inverse of a Rational Function · medium

Solution

Worked example 3: Constructing Transformed Functions · hard

Solution

Try these IB Maths AI HL composite, inverse and transformations questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 7 marks · Paper 1

Revise composite, inverse and transformations for the exam

FAQ

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