To determine the inverse of the function [tex]\( f(x) = \frac{z+2}{7} \)[/tex], let's follow a systematic approach:
1. Rewrite the function with [tex]\( y \)[/tex] instead of [tex]\( f(x) \)[/tex]:
[tex]\[
y = \frac{z+2}{7}
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse. If [tex]\( y = f(x) \)[/tex], we swap them to get [tex]\( x = f^{-1}(y) \)[/tex]:
[tex]\[
x = \frac{y+2}{7}
\][/tex]
3. Solve for [tex]\( y \)[/tex] to express the inverse function:
[tex]\[
x = \frac{y+2}{7}
\][/tex]
Multiply both sides by 7 to isolate [tex]\( y + 2 \)[/tex]:
[tex]\[
7x = y + 2
\][/tex]
Subtract 2 from both sides:
[tex]\[
7x - 2 = y
\][/tex]
Thus, the inverse function is:
[tex]\[
y = 7x - 2
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{A \text{. } p(x) = 7x - 2}
\][/tex]
