Functions, graphs, composites and inverses

Maths: Analysis and Approaches · Functions. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A function is a rule that gives exactly one output for each input. The domain is the set of inputs allowed and the range is the set of outputs that come out. Inputs are ruled out if they need a square root of a negative number or division by zero.

A composite function applies one function and then another. (f ∘ g)(x) means f(g(x)): g acts first, then f. The order matters.

The inverse f⁻¹ undoes f. It exists only if f is one-to-one, so sometimes the domain must be restricted first. The graph of f⁻¹ is the graph of f reflected in the line y = x.

Rules to remember

(f ∘ g)(x) = f(g(x)): work from the inside out; in general f ∘ g ≠ g ∘ f
To find f⁻¹: write y = f(x), swap x and y, then make y the subject
(f ∘ f⁻¹)(x) = (f⁻¹ ∘ f)(x) = x, the identity function
Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f
√(expression) needs expression ≥ 0; 1/(expression) needs expression ≠ 0
Zeros: solve f(x) = 0. y-intercept: find f(0)
Even function, f(−x) = f(x): graph symmetric about the y-axis

Common mistake

Doing a composite in the wrong order. In (f ∘ g)(x) the function written nearest to x, g, is applied first.

Worked example

f(x) = (x − 1)² + 2 for x ≥ 1. Find f⁻¹(x) and its domain.

Write y = (x − 1)² + 2 and swap: x = (y − 1)² + 2.
(y − 1)² = x − 2, so y − 1 = √(x − 2). The positive root is taken because y ≥ 1.
The range of f is y ≥ 2, so the domain of f⁻¹ is x ≥ 2.
Check: f(3) = 4 + 2 = 6 and f⁻¹(6) = 1 + √4 = 3.

Answer: f⁻¹(x) = 1 + √(x − 2), with domain x ≥ 2

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