To find the inverse of the function [tex]\( f(x) = \frac{1}{4}x - 12 \)[/tex], we follow these steps:
1. Express [tex]\( y = f(x) \)[/tex]:
[tex]\[
y = \frac{1}{4}x - 12
\][/tex]
2. Switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
x = \frac{1}{4}y - 12
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
First, isolate the term involving [tex]\( y \)[/tex]:
[tex]\[
x + 12 = \frac{1}{4}y
\][/tex]
Next, multiply both sides of the equation by 4 to clear the fraction:
[tex]\[
4(x + 12) = y
\][/tex]
Simplify the expression:
[tex]\[
y = 4x + 48
\][/tex]
4. Therefore, the inverse function [tex]\( h(x) \)[/tex] is:
[tex]\[
h(x) = 4x + 48
\][/tex]
This matches the fourth choice given in the problem. Thus, the inverse function is:
[tex]\[
h(x) = 4x + 48
\][/tex]
