To find the inverse of the function [tex]\( f(x) = \frac{1}{9}x + 2 \)[/tex], we need to follow a series of mathematical steps to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], and then solve for [tex]\( x \)[/tex]. Here’s a detailed step-by-step solution:
1. Rewrite the function [tex]\( f(x) \)[/tex] as [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{9}x + 2
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to begin finding the inverse:
[tex]\[
x = \frac{1}{9}y + 2
\][/tex]
3. Isolate [tex]\( y \)[/tex]:
- First, subtract 2 from both sides to move the constant term:
[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]
4. Simplify the equation:
[tex]\[
y = 9x - 18
\][/tex]
Therefore, the inverse function of [tex]\( f(x) = \frac{1}{9}x + 2 \)[/tex] is:
[tex]\[
h(x) = 9x - 18
\][/tex]
So, the correct answer from the given options is:
[tex]\[ h(x) = 9x - 18 \][/tex]
