To find the inverse of the function [tex]\( f(x) = 9x + 7 \)[/tex], we need to follow a series of steps. Here's a detailed, step-by-step solution:
1. Express the function in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 9x + 7 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
This step is fundamental in finding the inverse. By swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we prepare to solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
[tex]\[ x = 9y + 7 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
The goal is to isolate [tex]\( y \)[/tex]. We will perform algebraic operations to achieve this:
- First, subtract 7 from both sides of the equation to get:
[tex]\[ x - 7 = 9y \][/tex]
- Next, divide both sides by 9 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x - 7}{9} \][/tex]
4. Express the inverse function:
The expression [tex]\( y = \frac{x - 7}{9} \)[/tex] is the inverse function [tex]\( f^{-1}(x) \)[/tex]. We now write it as:
[tex]\[ f^{-1}(x) = \frac{x - 7}{9} \][/tex]
This can be further simplified to match one of the choices given in the problem:
[tex]\[ f^{-1}(x) = \frac{1}{9}x - \frac{7}{9} \][/tex]
Comparing this with the options provided:
A. [tex]\( f^{-1}(x) = \frac{7}{9} x - \frac{1}{9} \)[/tex]
B. [tex]\( f^{-1}(x) = \frac{1}{9} x - \frac{7}{9} \)[/tex] (Correct Answer)
C. [tex]\( f^{-1}(x) = 7 x + 9 \)[/tex]
D. [tex]\( f^{-1}(x) = -9 x - 7 \)[/tex]
Thus, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
