Q. 63 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 61–72, the function f is one-to-one. Find its inverse and check your answer. 

$f\left(x\right)=\frac{3x}{x+2}$

Step-by-Step Solution

Answer

The inverse of f  is $\frac{2x}{3-x}.$

Step 1. Given Information

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

We have to find its inverse and check the answer.

Step 2. Replace f(x) with y and interchange the variables x and y

So, $y=\frac{3x}{x+2}$

Interchange the variables

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

Step 3. Solve for y

$$\begin{gathered} x=\frac{3y}{y+2} \\ x\left(y+2\right)=3y \\ xy+2x=3y \\ 2x=3y-xy \\ 2x=y(3-x) \\ \frac{2x}{3-x}=y \end{gathered}$$

The inverse is $f^{-1}\left(x\right)=\frac{2x}{3-x}.$

Step 4. Check the answer

We can check the answer by $f^{-1}\left(f\left(x\right)\right) and f\left(f^{-1}\left(x\right)\right).$

So,

$$\begin{gathered} f^{-1}\left(f\left(x\right)\right)=f^{-1}\left(\frac{3x}{x+2}\right) \\ =\frac{2\left({\displaystyle \frac{3x}{x+2}}\right)}{3-\left({\displaystyle \frac{3x}{x+2}}\right)} \\ =\frac{6x}{6} \\ =x \end{gathered}$$

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

Now,

$$\begin{gathered} f\left(f^{-1}\left(x\right)\right)= f\left(\frac{2x}{3-x}\right) \\ =\frac{3\left({\displaystyle \frac{2x}{3-x}}\right)}{\left({\displaystyle \frac{2x}{3-x}}\right)+2} \\ =\frac{6x}{6} \\ =x \end{gathered}$$

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

Hence our result is right.