Q. 61 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

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

Step 1. Given Information

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

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, $y=\frac{2}{3+x}$

Interchange the variables

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

Step 3. Solve for y

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

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

Step 4. Check the answer

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

So,

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

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

Now,

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

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

Therefore, our answer is right.