Higher Maths · Expressions and Functions
Functions and Graphs | Higher Maths
Functions and Graphs covers the language and behaviour of functions you will use throughout Higher Maths. Composite functions, completing the square and graph transformations are tested on every Higher exam — typically as short questions on Paper 1 and longer multi-part questions on Paper 2.
Key results
Composite function: (f ∘ g)(x) = f(g(x))
Inverse function: if y = f(x) then x = f−1(y)
Completing the square: x2 + bx + c = (x + b/2)2 + c − (b/2)2
Translation: y = f(x − a) + b shifts graph by (a, b)
Reflection: y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis
Stretch: y = kf(x) stretches vertically by factor k; y = f(kx) stretches horizontally by 1/k
Worked example
Worked example — Composite functions
Problem: Given f(x) = 2x + 3 and g(x) = x2 − 1, find f(g(x)) and g(f(x)).
- Substitute g(x) into f.f(g(x)) = 2(x2 − 1) + 3 = 2x2 + 1
- Substitute f(x) into g.g(f(x)) = (2x + 3)2 − 1
- Expand and simplify.g(f(x)) = 4x2 + 12x + 9 − 1 = 4x2 + 12x + 8
Practice questions
Try these SQA-style questions. Tap "Show answer" to check your working.
Practice questions
- Express x2 − 6x + 11 in completed-square form.
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(x − 3)2 + 2 - If f(x) = 3x − 2, find f−1(x).
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f−1(x) = (x + 2)/3 - Sketch y = f(x) + 3 if y = f(x) passes through (0, 1) and (2, −3). Give the new key points.
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New key points: (0, 4) and (2, 0) — the graph is shifted up by 3. - Given f(x) = sin x and g(x) = 2x, write down (f ∘ g)(x) and (g ∘ f)(x).
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f(g(x)) = sin 2x ; g(f(x)) = 2 sin x - State the range of f(x) = (x − 1)2 + 4.
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f(x) ≥ 4
Common mistakes
Common mistakes & how to avoid them
- Confusing f ∘ g with g ∘ f — order matters: f(g(x)) means apply g first, then f.
- When completing the square, forgetting to subtract (b/2)2 after adding it inside the bracket.
- Sketching y = f(x − 2) by moving left instead of right. Inside the bracket, signs flip.
Frequently asked questions
Are graph transformations on the SQA formula sheet?
No. The behaviour of f(x ± a), −f(x), f(−x), kf(x) and f(kx) must be memorised.
How is "completing the square" tested in Higher?
It is used to find the maximum or minimum value of a quadratic, to derive the equation of a parabola, and as a setup for calculus problems.
What does the notation (f ∘ g) actually mean?
It means "f after g". You apply g first and then feed the result into f. Equivalent to f(g(x)).
Related Higher Maths topics
These topics often appear together in SQA exam questions.
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