What are composition of functions?

The composite function f g is defined by (f g)(x) = f(g(x)): apply g first, then f. Order matters, because in general f(g(x)) g(f(x)).

TASSpecialist MathematicsSyllabus dot point

How do we combine functions and reverse them while keeping domain and range correct?

Form composite and inverse functions, determining the correct domain and range of each.

Composition of functions, the existence of inverses, finding inverse functions and restricting domains, with attention to domain and range, for TCE Mathematics Specialised Unit 3.

Generated by Claude Opus 4.78 min answerUpdated 2026-05-30

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What this dot point is asking

This dot point opens the functions and sketching graphs topic. Before sketching the more exotic graphs in this strand, you need full control of how functions combine and reverse, and a disciplined habit of tracking domain and range at every step.

Composition of functions

The composite function $f \circ g$ is defined by $(f \circ g)(x) = f(g(x))$ : apply $g$ first, then $f$ . Order matters, because in general $f(g(x)) \ne g(f(x))$ .

For example, if $f(x) = \sqrt{x}$ (domain $x \ge 0$ ) and $g(x) = x - 4$ , then $f(g(x)) = \sqrt{x - 4}$ requires $x - 4 \ge 0$ , so the domain of the composite is $x \ge 4$ , narrower than the domain of $g$ alone.

Inverse functions

The inverse $f^{-1}$ undoes $f$ , so $f^{-1}(f(x)) = x$ . An inverse function exists only when $f$ is one-to-one, meaning no horizontal line crosses the graph more than once. If $f$ is many-to-one, you must restrict its domain to a piece on which it is one-to-one before an inverse exists.

Finding an inverse

To find an inverse: write $y = f(x)$ , swap $x$ and $y$ , then solve for $y$ . State the domain of the inverse, which equals the range of the original.

Inverse of a restricted quadratic

Worked example

Find the inverse of $f(x) = x^2 - 4$ for the restricted domain $x \ge 0$ , and state its domain and range.

On $x \ge 0$ the function is one-to-one, so an inverse exists. The range of $f$ here is $y \ge -4$ (the minimum is at $x = 0$ ).

Set $y = x^2 - 4$ , swap to get $x = y^2 - 4$ , then solve for $y$ :

y^2 = x + 4 \quad\Longrightarrow\quad y = \sqrt{x + 4},

taking the positive root because the original range of inputs was $x \ge 0$ , so the inverse outputs must be non-negative. Therefore

f^{-1}(x) = \sqrt{x + 4}, \qquad \text{domain } x \ge -4, \qquad \text{range } y \ge 0.

The domain of $f^{-1}$ matches the range of $f$ , and the range of $f^{-1}$ matches the restricted domain of $f$ , as expected.

Checking with composition

A reliable check is to confirm $f(f^{-1}(x)) = x$ . Here $f(f^{-1}(x)) = (\sqrt{x + 4})^2 - 4 = x + 4 - 4 = x$ on the stated domain. If the composition does not return $x$ , an algebra error or a domain mistake has crept in.

Why this matters

Composition and inverses underpin the rest of the sketching topic, where reciprocal, absolute value and rational functions are all built by transforming and combining simpler functions. Tracking domain and range carefully here prevents errors that would otherwise propagate into every later graph.