Functions summary

Key learning points

• $(fg)(x)=f(g(x))$ — apply the right-hand function first.
• Inverse: swap $x$ and $y$, then rearrange; reflect in $y=x$.
• Restrict the domain if a function is not one-to-one.
• State domains carefully for modulus and piecewise definitions.

Worked example 1 — Evaluate

$f(x)=2x-1$. Find $f(5)$.

Working

Substitute.

Worked example 2 — Composite

$f(x)=x+1$, $g(x)=2x$. Find $f(g(3))$.

Working

Apply $g$ then $f$.

Worked example 3 — Inverse

$f(x)=3x+2$. Find $f^{-1}(x)$.

Working

Swap and rearrange.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.