To find the inverse of the function [tex]\( f(x) = \frac{1}{9}x - 2 \)[/tex], follow these steps:
1. Express [tex]\( f(x) \)[/tex] as an equation with [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{9}x - 2
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to solve for the inverse function:
[tex]\[
x = \frac{1}{9}y - 2
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[
x + 2 = \frac{1}{9}y
\][/tex]
4. Multiply both sides by 9 to isolate [tex]\( y \)[/tex]:
[tex]\[
9(x + 2) = y
\][/tex]
5. Simplify the equation:
[tex]\[
y = 9x + 18
\][/tex]
So, the inverse function is:
[tex]\[
f^{-1}(x) = 9x + 18
\][/tex]
Therefore, the correct answer is:
[tex]\[
f^{-1}(x) = 9x + 18
\][/tex]
