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Q. 83

Expert-verified
Found in: Page 270

### Precalculus Enhanced with Graphing Utilities

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

# Find the inverse of the linear function$f\left(x\right)=mx+b\phantom{\rule{0ex}{0ex}}m\ne 0$

The inverse of the function$f\left(x\right)=mx+b\phantom{\rule{0ex}{0ex}}m\ne 0$ is ${f}^{-1}\left(x\right)=\frac{x-b}{m}\phantom{\rule{0ex}{0ex}}m\ne 0$

See the step by step solution

## Step 1. Given data

The given function is

$f\left(x\right)=mx+b\phantom{\rule{0ex}{0ex}}m\ne 0$

## Step 2. interchanging variables

Replace $f\left(x\right)$ with y and interchange x and y

role="math" localid="1646177714686" $f\left(x\right)=mx+b\phantom{\rule{0ex}{0ex}}y=mx+b\phantom{\rule{0ex}{0ex}}x=my+b$

## Step 3. The inverse of the function

Solve the equation for y

$x=my+b\phantom{\rule{0ex}{0ex}}x-b=my\phantom{\rule{0ex}{0ex}}\frac{x-b}{m}=y\phantom{\rule{0ex}{0ex}}y=\frac{x-b}{m}$

replace y with ${f}^{-1}\left(x\right)$

role="math" localid="1646177950266" ${f}^{-1}\left(x\right)=\frac{x-b}{m}\phantom{\rule{0ex}{0ex}}m\ne 0$

## Step 4. Verification

Determine $f\left({f}^{-1}\left(x\right)\right)$

$f\left({f}^{-1}\left(x\right)\right)=f\left(\frac{x-b}{m}\right)\phantom{\rule{0ex}{0ex}}f\left({f}^{-1}\left(x\right)\right)=m\left(\frac{x-b}{m}\right)+b\phantom{\rule{0ex}{0ex}}f\left({f}^{-1}\left(x\right)\right)=x-b+b\phantom{\rule{0ex}{0ex}}f\left({f}^{-1}\left(x\right)\right)=x$

Determine${f}^{-1}\left(f\left(x\right)\right)$

${f}^{-1}\left(f\left(x\right)\right)={f}^{-1}\left(mx+b\right)\phantom{\rule{0ex}{0ex}}{f}^{-1}\left(f\left(x\right)\right)=\frac{\left(mx+b\right)-b}{m}\phantom{\rule{0ex}{0ex}}{f}^{-1}\left(f\left(x\right)\right)=\frac{mx}{m}\phantom{\rule{0ex}{0ex}}{f}^{-1}\left(f\left(x\right)\right)=x$

${f}^{-1}\left(f\left(x\right)\right)=x&f\left({f}^{-1}\left(x\right)\right)=x$so inverse function is correct