To find the inverse of the function [tex]\( f(x) = \frac{1}{9}x + 2 \)[/tex], we need to follow these 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] to start finding the inverse function:
[tex]\[
x = \frac{1}{9}y + 2
\][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
First, we isolate the term with [tex]\( y \)[/tex]:
[tex]\[
x - 2 = \frac{1}{9}y
\][/tex]
Next, to get [tex]\( y \)[/tex] by itself, multiply both sides by 9:
[tex]\[
9(x - 2) = y
\][/tex]
This simplifies to:
[tex]\[
y = 9x - 18
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = 9x - 18
\][/tex]
Now, let's match this with the given options:
- [tex]\( h(x) = 18x - 2 \)[/tex]
- [tex]\( h(x) = 9x - 18 \)[/tex]
- [tex]\( h(x) = 9x + 18 \)[/tex]
- [tex]\( h(x) = 18x + 2 \)[/tex]
The correct option is:
[tex]\[
h(x) = 9x - 18
\][/tex]
