Describing Inverses of Functions

Before we can find the inverse of the given function, we need to replace $f(x)$ with $y.$ $$\begin{array}{c}f(x)=\text{-}2x+5\iff y=\text{-}2x+5\end{array}$$ To algebraically determine the inverse of the given function, we exchange $x$ and $y,$ and solve for $y.$ $$\begin{array}{c}y=\text{-}2x+5\stackrel{\text{switch}}{⟶}x=\text{-}2y+5\end{array}$$ The result of isolating $y$ in the new equation will be the inverse of the given function. Now, we will replace $y$ with $f^{\text{-}1}(x)$ to complete the operation. $$\begin{array}{c}y=\frac{5-x}{2}\iff f^{\text{-}1}(x)=\frac{5-x}{2}\end{array}$$

Graphing the Function

Because the given function is in slope-intercept form, we can graph it using its slope and $y\text{-}$intercept to graph it.

Graphing the Inverse of the Function

To graph the inverse linear function, let's write it in slope-intercept form. $$\begin{array}{c}{f}^{\text{-}1}(x)=\frac{5-x}{2}\iff f^{\text{-}1}(x)=\text{-}\frac{1}{2}x+\frac{5}{2}\end{array}$$ Now, let's add it to the graph of the given function so that we can more easily see how it is a reflection on the line $y=x.$