To find the y-intercept of the inverse function \( f^{-1}(x) \), we need to follow several steps in transforming and solving the given function. Here’s the detailed process:
1. Given Function:
The function is given as:
[tex]\[
f(x) = \frac{3}{4} x + 12
\][/tex]
2. Express as \( y = f(x) \):
Rewrite the function with \( y \):
[tex]\[
y = \frac{3}{4} x + 12
\][/tex]
3. Swap \( x \) and \( y \):
To find the inverse function, switch \( x \) and \( y \):
[tex]\[
x = \frac{3}{4} y + 12
\][/tex]
4. Solve for \( y \):
Isolate \( y \) to express the inverse function:
- Subtract 12 from both sides:
[tex]\[
x - 12 = \frac{3}{4} y
\][/tex]
- Multiply both sides by \( \frac{4}{3} \):
[tex]\[
y = \frac{4}{3} (x - 12)
\][/tex]
So, the inverse function is:
[tex]\[
f^{-1}(x) = \frac{4}{3} (x - 12)
\][/tex]
5. Find the \( y \)-intercept of the inverse function:
The y-intercept occurs when \( x = 0 \):
- Substitute \( x = 0 \) into the inverse function:
[tex]\[
f^{-1}(0) = \frac{4}{3} (0 - 12)
\][/tex]
[tex]\[
f^{-1}(0) = \frac{4}{3} \cdot (-12)
\][/tex]
[tex]\[
f^{-1}(0) = -16
\][/tex]
The y-intercept of the inverse function \( f^{-1}(x) \) is \( -16 \). Thus, the correct answer is:
[tex]\[
\boxed{-16}
\][/tex]
