§-Quick questions

NSWMaths AdvancedYear 12: Functions

Quick questions on Composite and inverse functions: existence, formulas, domains and graphs

6short 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

f \circ g

is the function

(f \circ g)(x) = f(g(x))

: apply

g

first, then

f

to the result.

What is finding an inverse algebraically?

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The inverse

f^{-1}

"undoes"

f

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

for

x \in \text{dom}(f)

and

f(f^{-1}(y)) = y

for

y \in \text{range}(f)

What is the inverse graph?

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The graph of

y = f^{-1}(x)

is the reflection of the graph of

y = f(x)

in the line

y = x

. Point

(a, b)

f

corresponds to

(b, a)

f^{-1}

. Horizontal asymptotes of

f

become vertical asymptotes of

f^{-1}

and vice versa.

What is see the inverse appear as a reflection, stage by stage?

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Take

f(x) = 2x - 3

(whose inverse is

f^{-1}(x) = \frac{x + 3}{2}

, found below). The four panels build the inverse purely by reflecting in

y = x

, with no algebra, so you can see why swapping coordinates is the same as mirroring.

What is restricting the domain?

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For a non-one-to-one function

f

, choose a domain on which

f

is one-to-one, then invert. Different restrictions give different inverses.

What is wrong domain for the inverse?

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The domain of

f^{-1}

equals the range of

f

. For

f(x) = e^x

with range

(0, \infty)

, the inverse

\ln x

has domain

(0, \infty)

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