Inverse functions

​​In a nutshell

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 input $$y$$​ and give the output $$x$$​.

Calculating inverse functions

The inverse of any given function can be calculated by two means: algebraically and graphically. The domain of $$f(x)$$​ is the range of $$f^{-1}(x)$$​ and vice versa.

Algebraically

To find the inverse of a function, rearrange the equation for $$x$$ and then swap the two variables $$x$$​ and $$y$$.

​$$x\implies f(x)\implies y$$​​

​$$y\implies f^{-1}(x)\implies x$$​​

Example 1

Find the inverse function of $$f(x)=6x-4$$:

To find the inverse function of the equation $$y=6x-4$$ simply rearrange for $$x$$ and swap the variables:

$$\begin{aligned}y &= 6x-4 \\y+4 &= 6x \\\dfrac{y+4}{6} &= x\end{aligned}$$​​​​

Therefore, the inverse function is $$\underline{f^{-1}(x)=\cfrac{x+4}{6}}$$​

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Graphically

To represent the inverse of a function, reflect the graph in the line $$y=x$$​.

Example 2

For the function $$f(x)=x^2$$, sketch the function $$y=f(x)$$ and the inverse function $$y=f^{-1}(x)$$ on the same set of axes.

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