To find the inverse of the function [tex]\( f(x) = \frac{1}{4}x - 12 \)[/tex], we will go through the following steps:
1. Set [tex]\( f(x) = y \)[/tex] and rewrite the function in terms of [tex]\( y \)[/tex]:
Given [tex]\( f(x) = \frac{1}{4}x - 12 \)[/tex], let [tex]\( y = \frac{1}{4}x - 12 \)[/tex].
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{4}x - 12
\][/tex]
To isolate [tex]\( x \)[/tex], first add 12 to both sides:
[tex]\[
y + 12 = \frac{1}{4}x
\][/tex]
Then multiply both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[
4(y + 12) = x
\][/tex]
Simplify the equation:
[tex]\[
x = 4y + 48
\][/tex]
3. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[
h(x) = 4x + 48
\][/tex]
So the inverse function [tex]\( h(x) \)[/tex] of [tex]\( f(x) = \frac{1}{4}x - 12 \)[/tex] is:
[tex]\[
h(x) = 4x + 48
\][/tex]
The correct answer is:
[tex]\[
h(x) = 4x + 48
\][/tex]
