Suggested languages for you:

Americas

Europe

Q. 68

Expert-verified
Found in: Page 269

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In Problems 61–72, the function f is one-to-one. Find its inverse and check your answer. $f\left(x\right)=\frac{2x-3}{x+4}$

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

See the step by step solution

## Step 1. Given Information

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

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

Interchange the variables

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

## Step 3. Solve for y

$x=\frac{2y-3}{y+4}\phantom{\rule{0ex}{0ex}}x\left(y+4\right)=2y-3\phantom{\rule{0ex}{0ex}}xy+4x=2y-3\phantom{\rule{0ex}{0ex}}4x+3=2y-xy\phantom{\rule{0ex}{0ex}}4x+3=y\left(2-x\right)\phantom{\rule{0ex}{0ex}}\frac{4x+3}{2-x}=y$

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

## 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,

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

Now,

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

Hence our result is right.