Inverse Functions

Video lesson · from 59:022 worked examples

One lesson video covers all of Functions & Graphs, so it opens at 59:02 for this topic — not from the beginning.

Inverse Functions 'reverse' each other.

If a function is defined on a suitable domain, then it will have an inverse function.

The domain of an inverse function is the range of the original function and vice versa.

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

x = 5

f(x) = 3x - 1

f(5) =

g(x) = \frac{x+1}{3}

g(14) =

g(f(5)) =

f(x) = 3x - 1

f(x) =

g(x) = \frac{x+1}{3}

g(3x-1) =

g(f(x)) =

Swap, then rearrange: write $y = f(x)$ , swap $x$ and $y$ , then make $y$ the subject. Rearranging without swapping is the most common failure.
Write the answer as $f^{-1}(x)$ : leaving it as $y = \ldots$ does not finish the question.
$f^{-1}$ is not a reciprocal: it does not mean $\frac{1}{f(x)}$ . The notation is unfortunate but the meaning is fixed.
An inverse needs a one-to-one function: some functions only have one if the domain is restricted first. Say so if the question calls for it.
Check by composing: $f(f^{-1}(x))$ should give $x$ . It is a fast, complete verification.

Example 1

Example 1

$f(x) = \frac{2}{1-x}$ and $g(x) = 1 - \frac{2}{x}$ where $x \neq 0, 1$ .

a) Find $f(g(x))$ .

b) State the connection between $f$ and $g$ .

\begin{aligned} f(g(x)) &= f\left(1 - \frac{2}{x}\right) \\ &= \frac{2}{1 - \left(1 - \frac{2}{x}\right)} \\ &= \frac{2}{\frac{2}{x}} \\ &= 2 \times \frac{x}{2} \\ &= x \end{aligned}

b) Since $f(g(x)) = x$ , $f$ and $g$ are inverse functions.

Example 2

Example 2

A function $f$ is defined, for all real numbers, by $f(x) = x^3 + 1$ .

Find a formula for its inverse $f^{-1}(x)$ .

Set $y = f(x)$ :

y = x^3 + 1

Change the subject to $x$ :

y - 1 = x^3

x = \sqrt[3]{y - 1}

Rewrite in terms of $x$ :

f^{-1}(x) = \sqrt[3]{x - 1}