What is the inverse of the function [tex]$f(x)=\frac{1}{9} x+2$[/tex]?

A. [tex]h(x)=18 x-2[/tex]
B. [tex]h(x)=9 x-18[/tex]
C. [tex]h(x)=9 x+18[/tex]
D. [tex]h(x)=18 x+2[/tex]



Answer :

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]