To find the inverse of the function [tex]\( f(x) = \frac{1}{9}x + 2 \)[/tex]:
1. Start with the function and let it equal [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] in terms of [tex]\( x \)[/tex]:
- First, isolate the term containing [tex]\( y \)[/tex]:
[tex]\[
x - 2 = \frac{1}{9}y
\][/tex]
- Next, multiply both sides by 9 to solve for [tex]\( y \)[/tex]:
[tex]\[
9(x - 2) = y
\][/tex]
- Simplify the right-hand side:
[tex]\[
y = 9x - 18
\][/tex]
4. Write the inverse function [tex]\( h(x) \)[/tex]:
[tex]\[
h(x) = 9x - 18
\][/tex]
Therefore, the inverse of the function [tex]\( f(x) = \frac{1}{9}x + 2 \)[/tex] is:
[tex]\[
h(x) = 9x - 18
\][/tex]
Hence, the correct answer is [tex]\( h(x) = 9x - 18 \)[/tex].
