Inverse functions

Warm-up

Three quick questions from earlier lessons.

Discover

$f(x) = x + 7$ adds 7. Which function undoes it, and gives back the number you started with?

$x - 7$
$\dfrac{1}{x + 7}$
$x + 7$

Key idea

The inverse function $f^{-1}$ undoes $f$ : it swaps inputs and outputs.

If

f(a) = b

, then

f^{-1}(b) = a

. To find

f^{-1}(x)

: write

y = f(x)

, solve for

x

, then rename. The graph of

f^{-1}

is the reflection of the graph of

f

across the line

y = x

.

$f^{-1}(x)$ is not $\dfrac{1}{f(x)}$ : the $-1$ is not an exponent here.

Worked example

$f(x) = 3x - 5$ . Find $f^{-1}(x)$ .

y = 3x - 5

Name the output y.

y + 5 = 3x

f subtracts 5 last, so undo that first: add 5.

x = \dfrac{y + 5}{3}

Then undo × 3: divide by 3.

f^{-1}(x) = \dfrac{x + 5}{3}

Rename the input x.

Your turn, guided

Your turn: $f(x) = 2x + 6$ .

y = 2x + 6

Name the output y.

y - 6 = 2x

Undo + 6 first.

f^{-1}(x) = \dfrac{x - 6}{2}

Divide by 2, then rename.

Practice ★

$f(x) = 7x + 9$ . Find $f^{-1}(x)$ .
$f(x) = 2x + 2$ . Find $f^{-1}(x)$ .
$f(x) = 7x + 5$ . Find $f^{-1}(x)$ .
$f(x) = 2x + 9$ . Find $f^{-1}(x)$ .

Watch out!

Practice ★★

$f(x) = 2x + 6$ . Find $f^{-1}(10)$ .
$f(x) = 2x + 9$ . Find $f^{-1}(11)$ .
$f(x) = 3x + 1$ . Find $f^{-1}(10)$ .

The point $(2, 7)$ is on the graph of $f$ . Tap the point that must be on the graph of $f^{-1}$ .
The point $(5, 7)$ is on the graph of $f$ . Tap the point that must be on the graph of $f^{-1}$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 7x + 9$ . Find $f^{-1}(x)$ .
$f(x) = 2x + 2$ . Find $f^{-1}(x)$ .
$f(x) = 7x + 5$ . Find $f^{-1}(x)$ .

Summary card

Inverse functions

Write $y = f(x)$ .
Solve for $x$ , undoing the last operation first.
Rename: $f^{-1}(x)$ . Check that $f(f^{-1}(x)) = x$ .

Watch out: $f^{-1}(x)$ is not $\dfrac{1}{f(x)}$ .

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