To find the inverse of the function [tex]\( f(x) = \frac{1}{9}x + 2 \)[/tex], we need to follow a few steps. Here’s a detailed, step-by-step method to determine the inverse:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{9}x + 2 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
This step is crucial because finding the inverse function essentially means expressing [tex]\( x \)[/tex] as a function of [tex]\( y \)[/tex].
[tex]\[ x = \frac{1}{9}y + 2 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
- Start by isolating the term involving [tex]\( y \)[/tex].
[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. Simplify the expression:
[tex]\[ y = 9(x - 2) \][/tex]
[tex]\[ y = 9x - 18 \][/tex]
So the inverse function is:
[tex]\[ h(x) = 9x - 18 \][/tex]
Thus, the correct option is:
[tex]\[ h(x) = 9x - 18 \][/tex]
