IB Maths AI SL · Unit 2: Functions
IB Maths AI SL Functions: Domain, Range & Inverses Questions
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 29 questions
- Paper 1: 22
- Paper 2: 7
- 4 easy
- 9 medium
- 7 hard
- 8 very hard
- 1 starter
- 5 worked examples
Practise Functions: Domain, Range & Inverses questions →
AI SL formula booklet
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 (20)
- Algebraic Inverse Functions (6)
- Inverse Function Properties (3)
Functions: Domain, Range & Inverses worked examples
Worked example 1: Inverse of a linear function · easy
Let $f(x) = 4x - 7$. Find an algebraic expression for the inverse function $f^{-1}(x)$.
1. Replace $f(x)$ with $y$: $y = 4x - 7$.
2. Swap $x$ and $y$: $x = 4y - 7$.
3. Add $7$: $x + 7 = 4y$.
4. Divide by $4$: $y = \frac{x + 7}{4}$.
5. State: $\mathbf{f^{-1}(x) = \frac{x + 7}{4}}$.
Examiner tip: Write your final answer using the correct notation $f^{-1}(x) = \ldots$ — leaving it as $y = \ldots$ or $x = \ldots$ typically loses the last accuracy mark.
Worked example 2: Asymptotes, domain and range of a rational function · medium
Consider $g(x) = \frac{3x + 2}{x - 5}$. State the equations of its vertical and horizontal asymptotes, and hence state the domain and range of $g(x)$.
1. Vertical asymptote: denominator zero: $x - 5 = 0 \implies x = 5$.
2. Horizontal asymptote: ratio of leading coefficients: $y = \frac{3}{1} = 3$.
3. Domain: exclude $x = 5$.
4. Range: exclude $y = 3$.
5. State: domain $\mathbf{x \in \mathbb{R},\ x \neq 5}$; range $\mathbf{y \in \mathbb{R},\ y \neq 3}$.
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 $x \ge 2$. 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 $\mathbf{y \ge 2}$.
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 $f(x) = \sqrt{x + 7}$.
1. Recall: for a real square root, the radicand must be $\ge 0$.
2. Set up: $x + 7 \ge 0$.
3. Subtract $7$: $x \ge -7$.
4. State: $\mathbf{x \ge -7}$ (i.e., $[-7,\ \infty)$).
Examiner tip: Radicals require the radicand to be $\ge 0$, NOT strictly $> 0$. $\sqrt{0} = 0$ is perfectly valid.
Worked example 5: Range of a quadratic function · medium
Find the exact range of $g(x) = x^2 - 6x + 10$.
1. Recognise $a = 1 > 0$: upward parabola, so the range is bounded below by the vertex $y$-value.
2. Vertex $x$: $x = \frac{-b}{2a} = \frac{6}{2} = 3$.
3. Substitute: $g(3) = 9 - 18 + 10$.
4. Evaluate: $g(3) = 1$.
5. State: $\mathbf{y \ge 1}$.
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.
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 \(y = 2x - 1\), 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.
Question 2 · medium · 5 marks · Paper 1
The graph of the semi-circle \(f(x) = \sqrt{16-x^2}\) is plotted below.
State the maximal domain of \(f(x)\).
State the range of \(f(x)\).
Question 3 · hard · 6 marks · Paper 2
A ball is thrown upwards from a balcony. Its height \(h(t)\) in metres after \(t\) seconds is modelled by \(h(t) = -5t^2 + 10t + 15\). The flight path is graphed below.
Find the exact time \(t\) when the ball hits the ground, and state the contextual domain.
Find the maximum height reached, and state the contextual range.
All 29 functions: domain, range & inverses questions with mark schemes →
FAQ
How many IB Maths AI SL functions: domain, range & inverses questions are there?
There are 29 exam-style functions: domain, range & inverses questions in the AI SL question bank (Paper 1: 22 · Paper 2: 7), graded 4 easy, 9 medium, 7 hard, 8 very hard, 1 starter. 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: 22 · 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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