To find the inverse of the function [tex]\( f(x) = \frac{1}{9} x + 2 \)[/tex], we need to follow a few steps:
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]:
[tex]\[ x = \frac{1}{9} y + 2 \][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
First, subtract 2 from both sides to isolate the term involving [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]
4. Define the inverse function [tex]\( h(x) \)[/tex]:
The inverse function [tex]\( h(x) \)[/tex] can now be written as:
[tex]\[ h(x) = 9(x - 2) \][/tex]
5. Simplify the expression:
Expand the expression inside the parentheses:
[tex]\[ h(x) = 9x - 18 \][/tex]
Thus, the inverse function is:
[tex]\[ h(x) = 9x - 18 \][/tex]
So, the correct answer is:
[tex]\[ h(x) = 9x - 18 \][/tex]
