To find the inverse of the function [tex]\( f(x) = \frac{1}{9}x + 2 \)[/tex], follow these detailed steps:
1. Rewrite the function as [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{9}x + 2 \][/tex]
2. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \frac{1}{9}y + 2 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
- Start by isolating [tex]\( y \)[/tex].
[tex]\[ x - 2 = \frac{1}{9}y \][/tex]
- Multiply both sides by 9 to solve for [tex]\( y \)[/tex]:
[tex]\[ 9(x - 2) = y \][/tex]
[tex]\[ y = 9x - 18 \][/tex]
4. Rewrite [tex]\( y \)[/tex] as 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].
So, the correct answer is:
[tex]\[ h(x) = 9x - 18 \][/tex]
