If [tex][tex]$f(x)=\frac{1}{9} x-2$[/tex][/tex], what is [tex][tex]$f^{-1}(x)$[/tex][/tex]?

A. [tex][tex]$f^{-1}(x)=9 x+18$[/tex][/tex]
B. [tex][tex]$f^{-1}(x)=\frac{1}{9} x+2$[/tex][/tex]
C. [tex][tex]$f^{-1}(x)=9 x+2$[/tex][/tex]
D. [tex][tex]$f^{-1}(x)=-2 x+\frac{1}{9}$[/tex][/tex]



Answer :

To find the inverse function [tex]\( f^{-1}(x) \)[/tex] of the given function [tex]\( f(x) = \frac{1}{9}x - 2 \)[/tex], follow these steps:

1. Express the function in terms of [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{9}x - 2 \][/tex]

2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = \frac{1}{9}y - 2 \][/tex]

3. Solve for [tex]\( y \)[/tex]:
First, isolate [tex]\( y \)[/tex] on one side of the equation:
[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. Rewrite [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = 9x + 18 \][/tex]

Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = 9x + 18 \][/tex]

Hence, the correct answer is:
[tex]\[ f^{-1}(x) = 9x + 18 \][/tex]