If [tex]$f(x)=9x-8[tex]$[/tex], which of the following is the inverse of [tex]$[/tex]f(x)$[/tex]?

A. [tex]$f^{-1}(x)=\frac{x-8}{9}$[/tex]
B. [tex]$f^{-1}(x)=\frac{x+9}{8}$[/tex]
C. [tex]$f^{-1}(x)=\frac{x+8}{9}$[/tex]
D. [tex]$f^{-1}(x)=\frac{x-9}{8}$[/tex]



Answer :

To find the inverse function of \( f(x) = 9x - 8 \), we follow these steps:

1. Start with the function \( f(x) = 9x - 8 \).

2. Replace \( f(x) \) with \( y \) for simplicity:
[tex]\[ y = 9x - 8 \][/tex]

3. To find the inverse, we need to solve this equation for \( x \). First, swap \( y \) and \( x \):
[tex]\[ x = 9y - 8 \][/tex]

4. Solve for \( y \):
[tex]\[ x = 9y - 8 \][/tex]
Add 8 to both sides:
[tex]\[ x + 8 = 9y \][/tex]
Divide both sides by 9 to isolate \( y \):
[tex]\[ y = \frac{x + 8}{9} \][/tex]

5. The inverse function \( f^{-1}(x) \) is therefore:
[tex]\[ f^{-1}(x) = \frac{x + 8}{9} \][/tex]

Checking the given options, we can see that option C corresponds to our derived inverse function:
[tex]\[ f^{-1}(x) = \frac{x + 8}{9} \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{f^{-1}(x) = \frac{x + 8}{9}} \][/tex]