Question

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

$f\left(x\right)=\frac{3x+4}{2x-3}$

Short answer

The inverse off is$\frac{4+3x}{2x-3}.$

Step-by-step solution

Step 1. Given Information 

The given function is$f\left(x\right)=\frac{3x+4}{2x-3}$

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,$y=\frac{3x+4}{2x-3}$

Interchange the variables

$x=\frac{3y+4}{2y-3}$

Step 3. Solve for y 

$$\begin{gathered} x=\frac{3y+4}{2y-3} \\ 2xy-3x=3y+4 \\ 2xy-3y=4+3x \\ y\left(2x-3\right)=4+3x \\ y=\frac{4+3x}{2x-3} \end{gathered}$$

The inverse is$f^{-1}\left(x\right)=\frac{4+3x}{2x-3}.$

Step 4. Check the answer 

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

So,

$$\begin{gathered} f^{-1}\left(f\left(x\right)\right)=f^{-1}\left(\frac{3x+4}{2x-3}\right) \\ =\frac{4+3\left({\displaystyle \frac{3x+4}{2x-3}}\right)}{2\left({\displaystyle \frac{3x+4}{2x-3}}\right)-3} \\ =\frac{17x}{17} \\ =x \end{gathered}$$

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

Now,

$$\begin{gathered} f\left(f^{-1}\left(x\right)\right)=f\left(\frac{4+3x}{2x-3}\right) \\ =\frac{3\left({\displaystyle \frac{4+3x}{2x-3}}\right)+4}{2\left({\displaystyle \frac{4+3x}{2x-3}}\right)-3} \\ =\frac{17x}{17} \\ =x \end{gathered}$$

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

Hence our answer is right.