Chapter 4.7, Problem 5CYU

To determine

To find: The inverse of the function $f\left(x\right)=-2x+7$ .

Answer to Problem 5CYU

The inverse of the function $f\left(x\right)=-2x+7$ is $f^{-1}\left(x\right)=-\frac{1}{2}x+\frac{7}{2}$ .

Explanation of Solution

Given:

The function is $f\left(x\right)=-2x+7$ .

Concept used:

To obtain the inverse of any function $f\left(x\right)=ax+b$ , following steps are used:

• Replace $f\left(x\right)$ with $y$ .
• Interchange $x$ and $y$ .
• Solve for $y$ .
• Again, replace $y$ with $f\left(x\right)$ .

Calculation:

Replace $f\left(x\right)$ with $y$ for the function $f\left(x\right)=-2x+7$ as follows:

$y=-2x+7$

Interchange $x$ and $y$ in the above expression.

$x=-2y+7$

Solve for $y$ in expression $x=-2y+7$ as follows:

$\begin{array}{c}x=-2y+7\\ 7-x=2y\\ y=\frac{7-x}{2}\\ y=-\frac{1}{2}x+\frac{7}{2}\end{array}$

Replace $y$ with $f^{-1}\left(x\right)$ in above equation to obtain the inverse of the function.

$f^{-1}\left(x\right)=-\frac{1}{2}x+\frac{7}{2}$

Thus, the inverse of the function $f\left(x\right)=-2x+7$ is $f^{-1}\left(x\right)=-\frac{1}{2}x+\frac{7}{2}$ .