Question

Find the inverse of the linear function$$\begin{gathered} f\left(x\right)=mx+b \\ m\ne 0 \end{gathered}$$

Short answer

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

Step-by-step solution

Step 1. Given data 

The given function is

$$\begin{gathered} f\left(x\right)=mx+b \\ m\ne 0 \end{gathered}$$

Step 2. interchanging variables 

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

role="math" localid="1646177714686" $$\begin{gathered} f\left(x\right)=mx+b \\ y=mx+b \\ x=my+b \end{gathered}$$

Step 3. The inverse of the function 

Solve the equation for y

$$\begin{gathered} x=my+b \\ x-b=my \\ \frac{x-b}{m}=y \\ y=\frac{x-b}{m} \end{gathered}$$

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

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

Step 4. Verification 

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

$$\begin{gathered} f\left(f^{-1}\right(x\left)\right)=f\left(\frac{x-b}{m}\right) \\ f\left(f^{-1}\right(x\left)\right)=m\left(\frac{x-b}{m}\right)+b \\ f\left(f^{-1}\right(x\left)\right)=x-b+b \\ f\left(f^{-1}\right(x\left)\right)=x \end{gathered}$$

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

$$\begin{gathered} f^{-1}\left(f\right(x\left)\right)=f^{-1}(mx+b) \\ {f}^{-1}\left(f\right(x\left)\right)=\frac{(mx+b)-b}{m} \\ {f}^{-1}\left(f\right(x\left)\right)=\frac{mx}{m} \\ {f}^{-1}\left(f\right(x\left)\right)=x \end{gathered}$$

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