Composite and inverse functions

In a nutshell

Functions take an input, perform an operation and then produce an output. Functions can be represented diagrammatically with a function machine or algebraically with $f(x)$ notation.

Once a function is written in function notation, it is possible to perform calculations to find the output, find composite functions and inverse functions. A composite function is a combination of two different functions, and an inverse function performs the inverse operation to the original function.

Function machines

Function machines are used to apply operations to an input to obtain an output. The input can be a number, a variable or even an expression. The function machine is represented in a diagram and can be written using $f(x)$ notation.

Example 1

Here is a function machine, find the output for the inputs $5$ , $x$ and $y+1$ .

$input \longrightarrow \fcolorbox{black}{white}{$\times 2$}\longrightarrow\fcolorbox{black}{white}{$+1$}\longrightarrow output$

$5 \longrightarrow \fcolorbox{black}{white}{$\times 2$}\longrightarrow\fcolorbox{black}{white}{$+1$}\longrightarrow \underline{11}$

$x \longrightarrow \fcolorbox{black}{white}{$\times 2$}\longrightarrow\fcolorbox{black}{white}{$+1$}\longrightarrow \underline{2x+1}$

$y+1 \longrightarrow \fcolorbox{black}{white}{$\times 2$}\longrightarrow\fcolorbox{black}{white}{$+1$}\longrightarrow 2(y+1)+1 = \underline{2y+3}$

Example 2

Write the function machine using $f(x)$ notation.

$x \longrightarrow \fcolorbox{black}{white}{$-2 $} \longrightarrow\fcolorbox{black}{white}{$square$}\longrightarrow y$

$\underline{f(x)=(x-2)^2}$

$f(x)$ notation

A function written algebraically in $f(x)$ notation can be used to calculate an output from an input. It is also possible to find the input given an output.

Example 3

Find the outputs for the function $f(x)=x^2-3$  with inputs $2, w$ and $y+1$ .

$\begin{aligned}f(x)&=x^2-3 \\f(2) &= 2^2-3 = \underline{1} \\f(w)&=\underline{w^2-3} \\f(y+1) &= (y+1)^2+3 = \underline{y^2+2y+4}\end{aligned}$

Example 4

For $f(x)= 2x-3$ , find the input when the output is $7$ .

Make the function equal to $7$ and then solve for $x$ .

$\begin{aligned}f(x)=2x-3&=7 \\2x-3&=7 \\2x&=10 \\\end{aligned}\\\underline{x=5}$

Composite functions

Composite functions consist of two functions combined together. The output of one function machine is used as an input for another function. The order of applying the functions matters. If an input is put through a function $f(x)$ then the output is put through $g(x)$ , the composite function is called $gf(x)$ .

$x \longrightarrow \fcolorbox{black}{white}{$f(x)$}\longrightarrow \fcolorbox{black}{white}{$g(x)$}\longrightarrow y$ is the same as $x \longrightarrow \fcolorbox{black}{white}{$gf(x)$}\longrightarrow y$

To find the composite function, substitute the inner function, which is $f(x)$ in the above example, into the outer function $g(x)$ .

Example 5

Find the composite function $fg(x)$ for the functions

$f(x) = x^2 \space and \space g(x)=x-5$

Substitute the inner function $g(x)$ into $f(x)$ to give

$\underline{fg(x)=(x-5)^2}$

Inverse functions

An inverse function $f^{-1}(x)$ performs the inverse operation to $f(x)$ . If $f(x)$ takes an input $x$ to calculate an output $y$ , then the inverse operation will take the value of $y$ as an input and give the output as $x$ .

$\begin{aligned}x \longrightarrow &\fcolorbox{black}{white}{$f(x)$} \longrightarrow y \\\\x \longleftarrow &\fcolorbox{black}{white}{$f^{-1}(x)$} \longleftarrow y \\\end{aligned}$

To find an inverse function, rearrange the function for $x$ and then swap $x$ and $y$ . Then rename the function to $f^{-1}(x)$ .

Example 6

Find the inverse function for

$f(x)=5x-3$

Write the equation as

$y=5x-3$

Rearrange for $x$ .

$\begin{aligned}y&=5x-3 \\y+3 &=5x \\\frac {y+3} 5 &= x \\x &= \frac {y+3} 5\end{aligned}$

Swap the $y$ for $x$ and rename the function to $f^{-1}(x)$ .

$\underline{f^{-1}(x)=\frac{x+3}{5}}$