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Functions and Graphs

Quick questions on Composite and inverse functions explained: H2 Mathematics Functions and Graphs

5short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What are composite functions?

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The composite

\mathrm{fg}

means "do

\mathrm{g}

first, then

\mathrm{f}

\mathrm{fg}(x) = \mathrm{f}(\mathrm{g}(x))

. The order matters, and in general

\mathrm{fg} \neq \mathrm{gf}

What are inverse functions?

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The inverse

\mathrm{f}^{-1}

undoes

\mathrm{f}

\mathrm{f}^{-1}(\mathrm{f}(x)) = x

. An inverse exists only if

\mathrm{f}

is one-to-one. To find it:

What is q1?

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Given

\mathrm{f}(x) = x^2

for

x \geq 0

and

\mathrm{g}(x) = x - 4

for

x \in \mathbb{R}

, find

\mathrm{gf}(x)

and

\mathrm{fg}(x)

. [3 marks]

What is q2?

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The function

\mathrm{f}

is defined by

\mathrm{f}(x) = 3 - 2x

for

x \in \mathbb{R}

. Find

\mathrm{f}^{-1}(x)

and verify

\mathrm{f}^{-1}\mathrm{f}(x) = x

. [3 marks]

What is q3?

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Explain the geometric relationship between the graphs of

\mathrm{f}

and

\mathrm{f}^{-1}

, and state where they can intersect. [2 marks]

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