To find the inverse function [tex]\( f^{-1}(x) \)[/tex] of the given function [tex]\( f(x) = \frac{1}{9}x - 2 \)[/tex], follow these steps:
1. Express the function in terms of [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{9}x - 2
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[
x = \frac{1}{9}y - 2
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
First, isolate [tex]\( y \)[/tex] on one side of the equation:
[tex]\[
x + 2 = \frac{1}{9}y
\][/tex]
Next, multiply both sides of the equation by 9 to solve for [tex]\( y \)[/tex]:
[tex]\[
9(x + 2) = y
\][/tex]
4. Rewrite [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = 9x + 18
\][/tex]
Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = 9x + 18
\][/tex]
Hence, the correct answer is:
[tex]\[
f^{-1}(x) = 9x + 18
\][/tex]
