Chapter 4.7, Problem 6CYU

To determine

To find: The inverse of the function $f\left(x\right)=\frac{2}{3}x+6$ .

Answer to Problem 6CYU

The inverse of the function $f\left(x\right)=\frac{2}{3}x+6$ is $f^{-1}\left(x\right)=\frac{3}{2}x-9$ .

Explanation of Solution

Given:

The function is $f\left(x\right)=\frac{2}{3}x+6$ .

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 $f\left(x\right)=\frac{2}{3}x+6$ as follows:

$y=\frac{2}{3}x+6$

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

$x=\frac{2}{3}y+6$

Solve for $y$ in expression $x=\frac{2}{3}y+6$ as follows:

$\begin{array}{c}x-6=\frac{2}{3}y\\ y=\frac{3}{2}\left(x-6\right)\\ y=\frac{3}{2}x-\frac{18}{2}\\ y=\frac{3}{2}x-9\end{array}$

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

$f^{-1}\left(x\right)=\frac{3}{2}x-9$

Thus, the inverse of the function $f\left(x\right)=\frac{2}{3}x+6$ is $f^{-1}\left(x\right)=\frac{3}{2}x-9$ .