To find the inverse of the function [tex]\( f(x) = \frac{1}{9} x + 2 \)[/tex], we need to follow a systematic process:
1. Start with the function [tex]\( y = f(x) \)[/tex]:
[tex]\[
y = \frac{1}{9} x + 2
\][/tex]
2. Swap the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This means we express [tex]\( x \)[/tex] as a function of [tex]\( y \)[/tex]:
[tex]\[
x = \frac{1}{9} y + 2
\][/tex]
3. Solve for [tex]\( y \)[/tex] to find the inverse function:
[tex]\[
x = \frac{1}{9} y + 2
\][/tex]
First, isolate [tex]\( y \)[/tex] on one side. Subtract 2 from both sides:
[tex]\[
x - 2 = \frac{1}{9} y
\][/tex]
Now, multiply both sides by 9 to solve for [tex]\( y \)[/tex]:
[tex]\[
9(x - 2) = y
\][/tex]
Simplify the expression:
[tex]\[
y = 9x - 18
\][/tex]
Therefore, the inverse function [tex]\( h(x) \)[/tex] is:
[tex]\[
h(x) = 9x - 18
\][/tex]
So the correct answer from the given options is:
[tex]\[
\boxed{h(x) = 9 x - 18}
\][/tex]
