Functions Topic Review summary

Key learning points

• Domain: allowed inputs; range: resulting outputs.
• Composite $(f\circ g)(x)=f(g(x))$; work inside-out.
• Inverse: swap variables then solve; reflect in $y=x$.
• One-to-one needed for an inverse on the whole domain.
• Modulus and piecewise definitions change domains and sketches.

Worked example 1 — Inverse

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

Working

Solve for $x$.

Worked example 2 — Composite

$f(x)=x^2$, $g(x)=x+1$. $fg(x)$?

Working

Apply $g$ first.

Worked example 3 — Domain: allowed inputs; range: resulting o

Apply this idea from the Topic Review: Domain: allowed inputs; range: resulting outputs.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Composite (f\\circ g)(x)=f(g(x)); work ins

Composite $(f\\circ g)(x)=f(g(x))$; work inside-out.

Working

Attempt without notes, then check the answers PDF.

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.