Relations and Functions: Notation, Composite and Inverse Functions
Domain, range and image, arrow diagrams and the vertical line test, functional notation, evaluating functions, composite functions and inverse functions.
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LESSON B8 · EXPLAINER
Relations and Functions: Notation, Composite and Inverse Functions
A function is a machine: put one number in, get exactly one number out. Composite and inverse functions are machines in a chain and machines running backwards.
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Step 2 · Read
Key notes
Copy these into your notebook in your own words — that is how they stick.
Relation
Any set of ordered pairs. Domain = inputs; range = outputs used; co-domain = set of possible outputs; image = output for a given input.
Function
Each element of the domain has exactly one image. One-to-one and many-to-one are functions; one-to-many is not.
Tests
Arrow diagram: one arrow per input. Graph: vertical line test.
Notation
f(x) = …; f : x → …; y = f(x). f(a) means substitute a for x.
Composite
fg(x) = f(g(x)) — g acts first. In general fg ≠ gf.
Inverse
Swap x and y and solve for y. f f⁻¹(x) = x. Graph of f⁻¹ is the reflection of f in y = x.
Solving f(x) = k
Set the rule equal to k and solve.
Inverse of composite
(fg)⁻¹ = g⁻¹ f⁻¹.
EXAM TIPS & COMMON MISTAKES
fg(x) means g FIRST, then f — read it right to left.
When evaluating f(−2), put the negative in brackets: 2(−2) + 3.
To check an inverse, compute f(f⁻¹(x)); the answer must be x.
A relation fails the function test if one input has TWO outputs — an input may not repeat with different images.
Play & practise
Games
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Step 3 · Practise
Paper 1 practice — multiple choice
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Step 4 · Apply
Paper 2 practice — structured questions
Write full answers, then reveal the mark scheme and tick the points you earned (18 marks in total). Show all your working — method marks count. Your answers are saved on this device.
CSEC Paper 02 style · Module 2 (Functions)
Question 1 [9 marks]
The functions f and g are defined by f(x) = 3x − 4 and g(x) = x².
(a) Evaluate f(5). [1]
MARK SCHEME — (a) (max 1)
(b) Find an expression for f⁻¹(x). [2]
MARK SCHEME — (b) (max 2)
(c) Show that gf(x) = (3x − 4)² and write gf(x) in expanded form. [3]
MARK SCHEME — (c) (max 3)
(d) Solve gf(x) = 25. [3]
MARK SCHEME — (d) (max 3)
CSEC Paper 02 style · Module 2 (Relations and inverse functions)
Question 2 [9 marks]
A relation is given by the set of ordered pairs {(1, 4), (2, 5), (3, 6), (3, 7)}.
(a) State the domain and the range of the relation. [2]
MARK SCHEME — (a) (max 2)
(b) Is the relation a function? Give a reason. [2]
MARK SCHEME — (b) (max 2)
(c) The function h is defined by h(x) = 5 − 2x. Find h(−1), h⁻¹(x), and show that h h⁻¹(3) = 3. [5]
MARK SCHEME — (c) (max 5)
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Sources
Where this topic appears in CSEC papers
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.
CSEC syllabus Module 2, SO 3.1–3.5 (relations and functions), 3.11 (composite functions), 3.12–3.13 (inverse functions), 3.14 (evaluating functions)
CSEC P1 May/June 2024 — range of f : x → x³ item; Paper 01 has four items on Relations, Functions and Graphs 1
CSEC P2 January 2023 — Question 8 (functions f, g and h: evaluate, find the inverse, composite value); P2 January 2026 — Question 8 (g(−1), inverse, composite functions)