To find the inverse function of \( f(x) = 9x - 8 \), we follow these steps:
1. Start with the function \( f(x) = 9x - 8 \).
2. Replace \( f(x) \) with \( y \) for simplicity:
[tex]\[
y = 9x - 8
\][/tex]
3. To find the inverse, we need to solve this equation for \( x \). First, swap \( y \) and \( x \):
[tex]\[
x = 9y - 8
\][/tex]
4. Solve for \( y \):
[tex]\[
x = 9y - 8
\][/tex]
Add 8 to both sides:
[tex]\[
x + 8 = 9y
\][/tex]
Divide both sides by 9 to isolate \( y \):
[tex]\[
y = \frac{x + 8}{9}
\][/tex]
5. The inverse function \( f^{-1}(x) \) is therefore:
[tex]\[
f^{-1}(x) = \frac{x + 8}{9}
\][/tex]
Checking the given options, we can see that option C corresponds to our derived inverse function:
[tex]\[
f^{-1}(x) = \frac{x + 8}{9}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{f^{-1}(x) = \frac{x + 8}{9}}
\][/tex]
