$h(x) = (x - 2)^2 + 4$ for the restricted domain $x \ge 2$. State the exact domain and range of $h^{-1}(x)$.

1. Original domain : $x \ge 2$ (given). 2. Original range : vertex at $(2, 4)$, opens up, so $y \ge 4$. 3. Recall : an inverse swaps domain and range. 4. Inverse domain : swap in original range $\implies x \ge 4$. 5. Inverse range : swap in original domain $\implies y \ge 2$. 6. State : domain $\mathbf{x \ge 4}$; range $\mathbf{y \ge 2}$.

Consider the natural logarithmic function $p(x) = \ln(x-3)$ . Write down the equation of the vertical asymptote. State the domain and range of $p(x)$ .

The vertical asymptote is where the argument of $\ln$ equals 0: $x - 3 = 0$ . (M1) $x = 3$ . A1 The argument must be strictly positive: $x - 3 > 0$ . (M1) Domain: $x > 3$ . A1 A logarithmic function takes all real values. Range: $y \in \mathbb{R}$ . A1

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IB Maths AI SL · Unit 2: Functions

IB Maths AI SL Functions: Domain, Range & Inverses Questions

Updated October 2026

Exam-style IB Maths AI SL functions: domain, range & inverses questions with worked solutions. Start with the 5 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.

28 questions
Paper 1: 21
Paper 2: 7
4 easy
9 medium
7 hard
8 very hard
5 worked examples

Practise Functions: Domain, Range & Inverses questions →

🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What's examined in AI SL functions: domain, range & inverses

The question bank covers these functions: domain, range & inverses question types (number of questions in brackets):

Domain and Range (19)
Algebraic Inverse Functions (6)
Inverse Function Properties (3)

Functions: Domain, Range & Inverses worked examples

Worked example 1: Inverse of a linear function · easy

Let $f (x) = 4 x - 7$ . Find an algebraic expression for the inverse function $f^{- 1} (x)$ .

Solution

5 steps

Examiner tip: Write your final answer using the correct notation

f^{- 1} (x) = \dots

— leaving it as

y = \dots

x = \dots

typically loses the last accuracy mark.

Worked example 2: Asymptotes, domain and range of a rational function · medium

Consider $g (x) = \frac{3 x + 2}{x - 5}$ . State the equations of its vertical and horizontal asymptotes, and hence state the domain and range of $g (x)$ .

Solution

5 steps · 1 to fill in

Examiner tip: Asymptotes are EQUATIONS of lines (

x = 5

y = 3

). Domain and range are SETS (

x \neq 5

y \neq 3

). Do not confuse the two forms.

Worked example 3: Domain and range of an inverse function · hard

$h (x) = (x - 2)^{2} + 4$ for the restricted domain $x \geq 2$ . State the exact domain and range of $h^{- 1} (x)$ .

Solution

6 steps

Examiner tip: You do NOT need to algebraically find the inverse equation to find its domain and range — just find them for the original function and swap.

Worked example 4: Domain of a square-root function · easy

Find the largest possible real domain of $f (x) = \sqrt{x + 7}$ .

Solution

4 steps

Examiner tip: Radicals require the radicand to be

\geq 0

, NOT strictly

> 0

\sqrt{0} = 0

is perfectly valid.

Worked example 5: Range of a quadratic function · medium

Find the exact range of $g (x) = x^{2} - 6 x + 10$ .

Solution

5 steps · 1 to fill in

Examiner tip: You can also just graph

g (x)

on the GDC and use the minimum tool to read off the smallest

y

-value directly.

Try these IB Maths AI SL functions: domain, range & inverses 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

Two relations are defined as follows.

Relation A: the set of points satisfying $x^{2} + y^{2} = 9$ (a circle with centre $(0, 0)$ and radius 3).
Relation B: the straight line $y = 2 x - 1$ , for all real values of $x$ .

Determine which of the two relations represents a function. Justify your answer using the vertical line test.
State the domain and range of Relation B.

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

Consider the natural logarithmic function $p (x) = \ln (x - 3)$ .

Write down the equation of the vertical asymptote.
State the domain and range of $p (x)$ .

Full worked solution Free, no sign-in

The vertical asymptote is where the argument of $\ln$ equals 0: $x - 3 = 0$ . (M1)
$x = 3$ . A1
The argument must be strictly positive: $x - 3 > 0$ . (M1)
Domain: $x > 3$ . A1
A logarithmic function takes all real values. Range: $y \in R$ . A1

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

Question 3 · hard · 5 marks · Paper 1

A taxi company charges $C (d) = 2.5 d + 4$ euros for a journey of $d$ km.

Find the cost of a $12$ km journey. [1 mark]
Find an expression for the inverse function $C^{- 1} (x)$ , and explain what $C^{- 1} (x)$ represents in this context. [3 marks]
A journey costs € $39$ . Use your answer to part (b) to find the length of the journey. [1 mark]

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AI SL functions: domain, range & inverses questions are there?

There are 28 exam-style functions: domain, range & inverses questions in the AI SL question bank (Paper 1: 21 · Paper 2: 7), graded 4 easy, 9 medium, 7 hard, 8 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is functions: domain, range & inverses on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 21 · Paper 2: 7. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL 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 lesson (slides and teacher notes): Domain, range & inverse functions.

IB Maths AI SL Functions: Domain, Range & Inverses Questions

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What's examined in AI SL functions: domain, range & inverses

Functions: Domain, Range & Inverses worked examples

Worked example 1: Inverse of a linear function · easy

Solution

Worked example 2: Asymptotes, domain and range of a rational function · medium

Solution

Worked example 3: Domain and range of an inverse function · hard

Solution

Worked example 4: Domain of a square-root function · easy

Solution

Worked example 5: Range of a quadratic function · medium

Solution

Try these IB Maths AI SL functions: domain, range & inverses 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 · 5 marks · Paper 1

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