Question

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

$f\left(x\right)=\frac{4}{2-x}$

Short answer

The inverse off is$\frac{2x-4}{x}.$

Step-by-step solution

Step 1. Given Information 

The given function is$f\left(x\right)=\frac{4}{2-x}$

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,$y=\frac{4}{2-x}$

Interchange the variables

$x=\frac{4}{2-y}$

Step 3. Solve for y 

$$\begin{gathered} x=\frac{4}{2-y} \\ x\left(2-y\right)=4 \\ 2x-xy=4 \\ 2x-4=xy \\ \frac{2x-4}{x}=y \end{gathered}$$

The inverse is$f^{-1}\left(x\right)=\frac{2x-4}{x}.$

Step 4. Check the answer 

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

So,

$$\begin{gathered} f^{-1}\left(f\left(x\right)\right)=f^{-1}\left(\frac{4}{2-x}\right) \\ =\frac{2\left({\displaystyle \frac{4}{2-x}}\right)-4}{\left({\displaystyle \frac{4}{2-x}}\right)} \\ =\frac{8-8+4x}{4} \\ =x \end{gathered}$$

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

Now,

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

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

Hence our answer is right.