To find the inverse of the function \( f(x) = \frac{1}{4} x - 12 \), we need to follow these steps:
1. Express \(y\) in terms of \(x\):
First, rewrite the function \(f(x)\) by replacing \(f(x)\) with \(y\):
[tex]\[
y = \frac{1}{4} x - 12
\][/tex]
2. Swap \(x\) and \(y\):
To find the inverse function, we swap \(x\) and \(y\):
[tex]\[
x = \frac{1}{4} y - 12
\][/tex]
3. Solve for \(y\):
We need to solve this equation for \(y\):
Multiply both sides by 4 to clear the fraction:
[tex]\[
4x = y - 48
\][/tex]
Add 48 to both sides to solve for \(y\):
[tex]\[
y = 4x + 48
\][/tex]
So, the inverse function \( f^{-1}(x) \) is:
[tex]\[
f^{-1}(x) = 4x + 48
\][/tex]
Among the given options:
- \(h(x) = 48x - 4\)
- \(h(x) = 48x + 4\)
- \(h(x) = 4x - 48\)
- \(h(x) = 4x + 48\)
The correct inverse function is:
[tex]\[
\boxed{h(x) = 4x + 48}
\][/tex]
