Inverse Functions

Reversing a function

A function can be imagined as a machine: an input enters, the function performs one or more operations, and an output emerges. An inverse function reverses this complete process. If $f$ maps an input of 5 to an output of 11, then $f^{-1}$ maps 11 back to 5:

$f(5)=11 \quad\Longleftrightarrow\quad f^{-1}(11)=5$

Pearson Edexcel expects the notation $f^{-1}(x)$ for an inverse function. This formal work, including finding the inverse of a linear function, is Higher-tier content.

The notation $f^{-1}(x)$ does not mean $\frac{1}{f(x)}$. It names the function that undoes .

Undoing operations in reverse order

Suppose

$f(x)=3x-4.$

The original function first multiplies by 3 and then subtracts 4. To reverse it, start with the output, add 4 and then divide by 3. The order must reverse because the subtraction happened after the multiplication:

$f^{-1}(x)=\frac{x+4}{3}.$

For example, $f(5)=3(5)-4=11$. Passing 11 through the inverse gives

$f^{-1}(11)=\frac{11+4}{3}=5,$

so the original input is recovered.

Finding an inverse algebraically

A reliable method is to write the output as $y$, rearrange to make the original input $x$ the subject, and then use inverse-function notation.

For example, let

$f(x)=4x-1.$

First write

$y=4x-1.$

Now make $x$ the subject:

$y+1=4x$

$x=\frac{y+1}{4}.$

This expression tells us how to obtain the old input from the old output. Replace the input label by $x$ and write the reversed rule as

$f^{-1}(x)=\frac{x+1}{4}.$

The letters are labels for the input and output; swapping them at the end expresses the reversed mapping as a new function of $x$.

Checking that the inverse works

A function followed by its inverse should return the starting value. Using $f(x)=4x-1$ and $f^{-1}(x)=\frac{x+1}{4}$, begin with any number :

$a\xrightarrow{f}4a-1\xrightarrow{f^{-1}}\frac{(4a-1)+1}{4}=a.$

The cancellation confirms that the rules undo each other. A numerical check can reveal an error, but this algebraic check shows that the result works for every permitted input.

Inputs, outputs and graphs

An inverse exchanges the roles of input and output. Therefore, if $(a,b)$ lies on the graph $y=f(x)$, then $(b,a)$ lies on $y=f^{-1}($. Geometrically, the two graphs are reflections of one another in the line .

This exchange also means that the domain of a function becomes the range of its inverse, while its range becomes the inverse's domain. A linear function with a non-zero gradient gives each output from exactly one input, so its inverse is also a function. By contrast, an unrestricted function such as $x^2$ gives the same output for two different inputs—for example, both 3 and produce 9—so reversing it would not give one unambiguous output unless its domain were restricted.