Q. 69 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

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

Step 1. Given Information

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

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

Interchange the variables

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

Step 3. Solve for y

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

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

Step 4. Check the answer

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

So,

$$\begin{gathered} f^{-1}\left(f\right(x\left)\right)=f^{-1}\left(\frac{2x+3}{x+2}\right) \\ =\frac{3-2\left({\displaystyle \frac{2x+3}{x+2}}\right)}{\left({\displaystyle \frac{2x+3}{x+2}}\right)-2} \\ =\frac{3-\left({\displaystyle \frac{4x+6}{x+2}}\right)}{{\displaystyle \frac{2x+3-2x-4}{x+2}}} \\ =\frac{{\displaystyle \frac{3x+6-4x-6}{x+2}}}{{\displaystyle \frac{2x+3-2x-4}{x+2}}} \\ =\frac{-x}{-1} \\ =x \end{gathered}$$

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

Now,

 $$\begin{gathered} f\left(f^{-1}\left(x\right)\right)=f\left(\frac{3-2x}{x-2}\right) \\ =\frac{2\left({\displaystyle \frac{3-2x}{x-2}}\right)+3}{\left({\displaystyle \frac{3-2x}{x-2}}\right)+2} \\ =\frac{\left({\displaystyle \frac{6-4x}{x-2}}\right)+3}{{\displaystyle \frac{3-2x+2x-4}{x-2}}} \\ =\frac{{\displaystyle \frac{6-4x+3x-6}{x-2}}}{{\displaystyle \frac{-1}{x-2}}} \\ =\frac{-x}{-1} \\ =x \end{gathered}$$

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

Therefore, our answer is right.