To find the inverse of the function [tex]\( f(x) = \frac{10}{9} x + 11 \)[/tex], we need to go through the following steps:
1. Rewrite the function as an equation involving [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
y = \frac{10}{9} x + 11
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
- First, subtract 11 from both sides:
[tex]\[
y - 11 = \frac{10}{9} x
\][/tex]
- Next, multiply both sides by [tex]\(\frac{9}{10}\)[/tex] to isolate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{9}{10} (y - 11)
\][/tex]
3. Simplify the expression:
[tex]\[
x = \frac{9}{10} y - \frac{9}{10} \cdot 11
\][/tex]
[tex]\[
x = \frac{9}{10} y - 9.9
\][/tex]
4. Rearrange the equation to express [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = \frac{9}{10} x - 9.9
\][/tex]
Simplify further:
[tex]\[
f^{-1}(x) = \frac{9 x - 99}{10}
\][/tex]
Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] is [tex]\(\boxed{\frac{9 x - 99}{10}}\)[/tex].
So, the correct answer is:
C. [tex]\( f^{-1}(x) = \frac{9 x - 99}{10} \)[/tex]
