Q. 70 - Precalculus Enhanced with Graphing Utilities

Question

the function f is one-to-one. Find its inverse and check your answer.

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

Step-by-Step Solution

Answer

The inverse of the function $f\left(x\right)=\frac{-3x-4}{x-2}$is $$\begin{gathered} f^{-1}\left(x\right)=\frac{2x-4}{x+3} \\ x\ne -3 \end{gathered}$$

Step 1. Given data

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

Step 2. interchanging variables

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

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

Step 3. The inverse of the function

Solve the equation for y

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

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

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

Step 4. Verification

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

$$\begin{gathered} f\left(f^{-1}\right(x\left)\right)=f\left(\frac{2x-4}{x+3}\right) \\ f\left(f^{-1}\right(x\left)\right)=\frac{-3\left(\frac{2x-4}{x+3}\right)-4}{\left(\frac{2x-4}{x+3}\right)-2} \\ f\left(f^{-1}\right(x\left)\right)=\frac{-3(2x-4)-4(x+3)}{\left(2x-4\right)-2(x+3)} \\ f\left(f^{-1}\right(x\left)\right)=\frac{-6x+12-4x-12}{2x-4-2x-6} \\ f\left(f^{-1}\right(x\left)\right)=\frac{-10x}{-10} \\ f\left(f^{-1}\right(x\left)\right)=x \end{gathered}$$

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

$$\begin{gathered} f^{-1}\left(f\right(x\left)\right)=f^{-1}\left(\frac{-3x-4}{x-2}\right) \\ {f}^{-1}\left(f\right(x\left)\right)=\frac{2\left(\frac{-3x-4}{x-2}\right)-4}{\left(\frac{-3x-4}{x-2}\right)+3} \\ {f}^{-1}\left(f\right(x\left)\right)=\frac{2(-3x-4)-4(x-2)}{(-3x-4)+3(x-2)} \\ {f}^{-1}\left(f\right(x\left)\right)=\frac{-6x-8-4x+8}{-3x-4+3x-6} \\ {f}^{-1}\left(f\right(x\left)\right)=\frac{-10x}{-10} \\ {f}^{-1}\left(f\right(x\left)\right)=x \end{gathered}$$

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