Question

show steps to solve.
6i find the inverse of the function $f(x) = \frac{1}{4}x - 6$ algebraically.
change f(x) to y

switch x and y.

solve for y

answer —— change y to $f^{-1}(x)$

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start by writing the function \( f(x)=\frac{1}{4}x - 6 \) as \( y=\frac{1}{4}x - 6 \).

Step2: Swap \( x \) and \( y \)

Interchange the roles of \( x \) and \( y \) to get \( x=\frac{1}{4}y - 6 \).

Step3: Solve for \( y \)

First, add 6 to both sides of the equation: \( x + 6=\frac{1}{4}y \). Then, multiply both sides by 4 to isolate \( y \): \( y = 4(x + 6)=4x+24 \).

Step4: Replace \( y \) with \( f^{-1}(x) \)

We now write the inverse function as \( f^{-1}(x)=4x + 24 \).

Answer:

The inverse of the function \( f(x)=\frac{1}{4}x - 6 \) is \( f^{-1}(x)=4x + 24 \)