Functions

Learn, then practise

The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.

Inverse functions, domain and range

Distinguish a function from a one-one function.
Find ranges, respecting unattained and excluded values.
Find and verify inverse rules with their domains and ranges.

A function gives exactly one output for each allowed input. One-one means distinct inputs give distinct outputs. For an inverse, solve \(y=f(x)\) for \(x\), then exchange variable names. Domain and range swap; \(f^{-1}\) does not mean \(\displaystyle \frac{1}{f}\).

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Composite functions and their domains

Use fg(x)=f(g(x)) consistently.
Distinguish full-domain composition from an explicitly restricted composition.
Retain intermediate restrictions after simplification.

We use \(fg(x)=f(g(x))\): apply \(g\) first. To form \(fg\) on all of \(D_g\), require \(R_g\subseteq D_f\). If this fails, a question may explicitly ask you to restrict the inner domain to \(\{x\in D_g:g(x)\in D_f\}\). Do not silently change a stated domain.

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Restricting a quadratic to obtain an inverse

Use a vertex to choose a maximal one-one branch.
Select the correct square-root sign.
Transfer strict or finite endpoint restrictions to the inverse.

A quadratic is one-one on either side of its vertex, but not across it. Choose the square-root sign from the ORIGINAL domain. An inverse function has one output, so do not leave \(\pm\) in its rule. A restricted endpoint need not be the vertex.

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A function and its inverse on a graph

Reflect complete graph features in y=x.
Preserve open endpoints, domains and ranges.
Find common points with justification; recognise decreasing counterexamples.

Reflection in \(y=x\) sends \((u,v)\) to \((v,u)\). Mark the mirror line on sketches and preserve open or closed endpoints. For a strictly increasing function, common points with its inverse are fixed points. This need not hold for a decreasing function.

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Transformations, stretches and coordinate rules

Describe translations, reflections and stretches accurately.
Use point mappings for combined transformations.
Transform full domains and ranges, including strict endpoints.

For \(y=A f(Bx+C)+D\), with \(B\ne0\), a point \((u,v)\) maps to \(\displaystyle (\frac{u- C}{B}, Av+ D)\). The horizontal scale factor is \(\displaystyle \frac{1}{|B|}\). A negative \(B\) reflects horizontally; a negative \(A\) reflects vertically. State the order when describing a sequence.

Read the full notes and 3 worked examples · Next: short practice