Question

In the following exercises, find the inverse of each function.

$f\left(x\right)=-2x+5$

Short answer

The inverse of the function is$f^{-1}\left(x\right)=\frac{5-x}{2}$

Step-by-step solution

Step 1. Given data

The given function is$f\left(x\right)=-2x+5$

Step 2. Substitute y for f(x)

Replace $f\left(x\right)$with $y$

role="math" localid="1644861539524" $$\begin{gathered} f\left(x\right)=-2x+5 \\ y=-2x+5 \end{gathered}$$

Step 3. Exchange x & y

Interchange variables$x\&y$role="math" localid="1644861582338" $$\begin{gathered} y=-2x+5 \\ x=-2y+5 \end{gathered}$$

Step 4. solve the equation for y

Subtract $5$from both sides

$$\begin{gathered} x-5=-2y+5-5 \\ x-5=-2y \end{gathered}$$

Divide both sides by $-2$

role="math" localid="1644864407879" $$\begin{gathered} \frac{x-5}{-2}=\frac{-2y}{-2} \\ \frac{5-x}{2}=y \end{gathered}$$

Step 5. Substitute f-1(x) for y

Replace $y$with $f^{-1}\left(x\right)$

role="math" localid="1644866422646" $$\begin{gathered} \frac{5-x}{2}=y \\ \frac{5-x}{2}=f^{-1}\left(x\right) \\ {f}^{-1}\left(x\right)=\frac{5-x}{2} \end{gathered}$$

Step 6. verification

Show $f^{-1}\left(f\right(x\left)\right)=x$

role="math" localid="1644866304794" $$\begin{gathered} f^{-1}(-2x+5)\stackrel{?}{=}x \\ \frac{5-(-2x+5)}{2}\stackrel{?}{=}x \\ \frac{5+2x-5}{2}\stackrel{?}{=}x \\ x=x \end{gathered}$$

Show $f\left(f^{-1}\right(x\left)\right)=x$

role="math" localid="1644866412914" $$\begin{gathered} f\left(\frac{5-x}{2}\right)\stackrel{?}{=}x \\ -2\left(\frac{5-x}{2}\right)+5\stackrel{?}{=}x \\ -5+x+5\stackrel{?}{=}x \\ x=x \end{gathered}$$