Composite Functions

A composite function applies one function to the output of another. The notation is the whole battle: $fg(x)$ means $g$ first, then $f$, read right to left, like the innermost bracket of $f(g(x))$.

Building a composite: brackets first

$f(x) = 3x - 2$, $g(x) = x^2 + 1$. Find $fg(x)$ and $gf(x)$. $fg(x) = f(g(x)) = 3(x^2 + 1) - 2 = 3x^2 + 1$ $gf(x) = g(f(x)) = (3x - 2)^2 + 1 = 9x^2 - 12x + 5$

The working habit that earns the method mark: write the substitution line with the inner function still bracketed. $3(x^2 + 1) - 2$, before expanding. Jumping straight to the expanded form risks the whole question on one slip, and shows the examiner nothing creditable if it goes wrong.

Note $fg \ne gf$ in general. If your two composites come out identical, recheck, it happens, but rarely in exam questions.

Solving $fg(x) = k$

Build the composite first, then solve the resulting equation:

$fg(x) = 7$: $3x^2 + 1 = 7 \to x^2 = 2 \to x = \pm\sqrt{2}$

Keep both roots unless the domain excludes one (then reject in writing, the rejection sentence is routinely a mark). For $f^2(x)$, 0606 means $ff(x)$, apply $f$ twice, not $[f(x)]^2$.

Domains of composites

The composite $fg$ only accepts $x$-values that $g$ accepts and whose outputs $f$ accepts. In practice 0606 tests this lightly, but if $g(x) = \sqrt{x}$ feeds $f$, the domain restriction $x \ge 0$ travels with it, and stating that is a mark when asked.

Common mistakes

• Order reversed: computing $gf$ when $fg$ was asked (the single most common error in scripts)
• Brackets dropped: $3x^2 + 1 - 2$ instead of $3(x^2 + 1) - 2$
• $f^2(x)$ treated as squaring instead of double application
• Solving $f(x)g(x) = k$ as if it were a composite, a product is not a composition

Composites combine naturally with inverse functions (e.g. solving $fg(x) = 5$ via $g(x) = f^{-1}(5)$), a favourite harder part. Full topic context: Functions notes.