# Composite and inverse functions

### 2) Inverse functions

An inverse function is a second function which undoes the work of the ﬁrst one. In this unit we describe two methods for ﬁnding inverse functions, and we also explain that the domain of a function may need to be restricted before an inverse function can exist.

Question 1

If $f(x) = 7x - 2$, find $f^{-1}(x)$

First, rearrange in terms of:
$y = 7x - 2$
$7x - 2 = y$
$7x = y + 2$
$x = \frac{y + 2}{7}$

Remember to change y back to x when you’re writing your answer:
$x = \frac{y + 2}{7}$, therefore $f^{-1}(x)=\frac{x + 2}{7}$

Question 2

Find the inverse of $f(x) = \frac{2x-3}{7}$.

To ﬁnd the inverse of f(x), we swap the x and y values which represent the inputs and outputs. To ﬁnd the inverse function we now make y the subject.

$y = \frac{2x-3}{7}$
$x = \frac{2y-3}{7}$
$7x + 3 = 2y$
$y = \frac{7x-3}{2}$

Therefore, $f^{-1}(x)=\frac{7x-3}{2}$

Now use the same method to solve this question.

(a) $f(x) =\frac{3}{4x+4}$ for $x > 1$
(b) $f(x) =\frac{x + 1}{x + 2}$ for $x > -2$

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