Question

show steps to solve.

1. 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: Change \( f(x) \) to \( y \)

Given the function \( f(x)=\frac{1}{4}x - 6 \), we replace \( f(x) \) with \( y \). So we get \( y=\frac{1}{4}x - 6 \).

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

Now we swap the roles of \( x \) and \( y \) in the equation \( y=\frac{1}{4}x - 6 \). This gives us \( x=\frac{1}{4}y - 6 \).

Step3: Solve for \( y \)

First, we add 6 to both sides of the equation \( x=\frac{1}{4}y - 6 \) to isolate the term with \( y \). So we have \( x + 6=\frac{1}{4}y \). Then, we multiply both sides by 4 to solve for \( y \). That is \( 4(x + 6)=y \), which simplifies to \( y = 4x+24 \).

Step4: Change \( y \) to \( f^{-1}(x) \)

We replace \( y \) with \( f^{-1}(x) \) to denote the inverse function. So \( f^{-1}(x)=4x + 24 \).

Answer:

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