To find the inverse of the function [tex]\( f(x) = \frac{x + 2}{7} \)[/tex], we need to follow these steps:
1. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = \frac{x + 2}{7}
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
This means replacing all instances of [tex]\( y \)[/tex] by [tex]\( x \)[/tex], and all instances of [tex]\( x \)[/tex] by [tex]\( y \)[/tex]:
[tex]\[
x = \frac{y + 2}{7}
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
To isolate [tex]\( y \)[/tex], start by multiplying both sides of the equation by 7:
[tex]\[
7x = y + 2
\][/tex]
Then, subtract 2 from both sides:
[tex]\[
y = 7x - 2
\][/tex]
So the inverse function is [tex]\( f^{-1}(x) = 7x - 2 \)[/tex].
From the given options:
- Option A: [tex]\(\frac{-x + 2}{7}\)[/tex] is not the correct inverse.
- Option B: [tex]\(\frac{7}{x + 2}\)[/tex] is not the correct inverse.
- Option C: [tex]\(7x - 2\)[/tex] corresponds to the correct inverse function we found.
- Option D: [tex]\(2x + 7\)[/tex] is not the correct inverse.
Thus, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
