Chapter 4.7, Problem 18PPS

To determine

To find: The inverse of the function $f\left(x\right)=\frac{2}{5}x+10$ .

Answer to Problem 18PPS

The inverse of the function $f\left(x\right)=\frac{2}{5}x+10$ is $f^{-1}\left(x\right)=\frac{5}{2}x-25$ .

Explanation of Solution

Given:

The function is $f\left(x\right)=\frac{2}{5}x+10$ .

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 $f\left(x\right)=\frac{2}{5}x+10$ as follows:

$y=\frac{2}{5}x+10$

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

$x=\frac{2}{5}y+10$

Solve for $y$ in expression $x=\frac{2}{5}y+10$ as follows:

$\begin{array}{c}x-10=\frac{2}{5}y\\ \frac{5}{2}\left(x-10\right)=y\\ y=\frac{5}{2}x-\frac{50}{2}\\ y=\frac{5}{2}x-25\end{array}$

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

$f^{-1}\left(x\right)=\frac{5}{2}x-25$

Thus, the inverse of the function $f\left(x\right)=\frac{2}{5}x+10$ is $f^{-1}\left(x\right)=\frac{5}{2}x-25$ .