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IB Maths AA SL · Unit 2: Functions

Functions: Domain, Range & Inverses (IB Maths AA SL): notes, exam questions and worked examples

Updated October 2026 · By Pete Bromfield

A function sends each input in its domain to exactly one output; the set of outputs is the range. A composite f(g(x)) applies g first, then f. An inverse f−1 undoes f, exists only when f is one-to-one, and its graph is the reflection of y=f(x) in y=x.

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Start with one question on composite functions. Stuck? Take a hint. No sign-in needed.

f(x)=2x+3 and g(x)=x2+1. Find (f∘g)(4).
Choose your answer · keys 1–4

Functions: Domain, Range & Inverses notes

Domain and range

Look for what is not allowed: a square root needs a non-negative inside, ln needs a positive argument, and a denominator cannot be zero. So f(x)=x−3 has domain x≥3 and range f(x)≥0. Find a range from the graph: sketch it over the domain and read off the lowest and highest y-values, including any asymptote the graph approaches but never reaches.

Composite functions

(f∘g)(x)=f(g(x)): work from the inside out. With f(x)=2x+1 and g(x)=x2, (f∘g)(x)=2x2+1 but (g∘f)(x)=(2x+1)2, so the order matters. The outputs of g must lie in the domain of f.

Inverse functions

To find f−1, write y=f(x), swap x and y, and make y the subject. For f(x)=3x−2, f−1(x)=x+23. Check: f(f−1(x))=x.

The domain of f−1 is the range of f. To find a single value such as f−1(7), you do not need the formula: solve f(x)=7, which gives x=3.

Graphs of inverse functions

Reflect y=f(x) in the line y=x: the point (a,b) becomes (b,a), so intercepts and asymptotes swap axes. A function that fails the horizontal line test has no inverse unless you restrict its domain, for example x≥0 for x2.

Key formulas

Composite; inverse Learn it
(f∘g)(x)=f(g(x));f(f−1(x))=x
Graph of f−1: reflect y=f(x) in y=x; domain of f−1 = range of f Learn it

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AA SL formulas →

Functions: Domain, Range & Inverses worked examples

Worked example 1: Finding an inverse function algebraically · easy

The function f is defined by f(x)=2x−4. Find an algebraic expression for f−1(x).

Solution

5 steps
Examiner tip: Always replace y with the formal inverse notation f−1(x) in your final answer to avoid losing the final accuracy mark.

Worked example 2: Composite functions and ranges · medium

Let f(x)=12x+2 and g(x)=x2−4x+3. Find a simplified expression for (f∘g)(x) and hence determine its exact range.

Solution

4 steps · 1 to fill in
Examiner tip: When finding the range of a composite quadratic, locating the vertex is essential because it represents the global minimum or maximum boundary.

Worked example 3: Inverse of a rational function · hard

Let f(x)=x+5x−1 for x≠1. Find f−1(x) and briefly explain the geometric significance.

Solution

6 steps
Examiner tip: For rational functions with variables in both numerator and denominator, grouping y terms on one side and factorising is the only reliable algebraic path forward.

Functions: Domain, Range & Inverses exam-style questions

3 questions from our AA SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Each has a full mark scheme in the practice area.

Question 1 · easy · 3 marks · Paper 1

A one-to-one function h is defined such that h(−2)=5.

  1. Write down the exact value of h−1(5).
  2. A new function is defined by k(x)=3h(x)+4. Find the exact value of k(−2).

Full worked solution Free, no sign-in

  1. Since h(−2)=5, the inverse gives h−1(5)=−2. A1

  2. Substitute h(−2)=5: k(−2)=3h(−2)+4=3(5)+4. M1
    k(−2)=19 A1

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

Question 2 · medium · 5 marks · Paper 2

Let f(x)=3x−2.

  1. Sketch the graph of f(x) for −2≤x≤2 on a set of axes. Clearly label the y-intercept and the horizontal asymptote.

  2. Write down the domain and range of f−1(x).

Attempt it and see the mark scheme →

Question 3 · hard · 7 marks · Paper 1

Let f(x)=x+4 and g(x)=x−2.

  1. Find the domain and range of f(x).

  2. Sketch the graphs of f(x) and g(x) on the same grid.

  3. Use algebra to find the exact coordinates of the point(s) of intersection.

Attempt it and see the mark scheme →

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Where students lose marks on functions: domain, range & inverses

Examiner's noteTransformations and inverses: incomplete descriptions and f⁻¹ notation

Where marks are lost: Candidates described only one of two transformations, gave a translation in the wrong direction, mishandled a vertical stretch (multiplying a function by a constant), misapplied log laws before transforming, and did not recognise f⁻¹ notation or that the domain of f⁻¹ is the range of f.

Examiners flagged this in 4 exam series (May 2024, November 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Functions: Domain, Range & Inverses on your calculator

A GDC is allowed on Paper 2, not on Paper 1. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Functions: Domain, Range & Inverses FAQ

What does f⁻¹(a) mean in an exam question?

It is the input that f sends to a. You can find it without the inverse formula by solving f(x) = a.

Does every function have an inverse?

No. Only one-to-one functions do: each output must come from exactly one input. A function such as x² has an inverse only after you restrict its domain, for example to x ≥ 0.

How many IB Maths AA SL function fundamentals questions are there?

There are 53 exam-style function fundamentals questions in the AA SL question bank (Paper 1: 43 · Paper 2: 10), graded 10 easy, 28 medium, 9 hard, 6 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is function fundamentals on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 43 · Paper 2: 10. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Next steps

More AA SL Unit 2: Equations of Straight Lines · Exponential and Logarithmic Functions · Graphing Functions · Rational and Reciprocal Functions · Transformations of Graphs

Teaching this topic? Free ready-to-teach lessons (slides and teacher notes): Function notation, domain & range · Composite functions (f∘g) · Inverse functions & their graphs.

Keep revising AA SL: AA SL revision notes · IB Maths AA SL past papers · AA SL common exam mistakes · AA SL syllabus checklist · AA SL functions knowledge organiser

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