Sure, let's break down the transformation of the function [tex]\( f(x) = \frac{1}{x} \)[/tex] step-by-step.
1. Horizontal Shift:
To shift the graph 2 units to the right, we replace [tex]\( x \)[/tex] with [tex]\( x - 2 \)[/tex]. Hence, our function becomes:
[tex]\[
f(x - 2) = \frac{1}{x - 2}
\][/tex]
2. Vertical Shift:
To shift the graph 7 units up, we add 7 to the function. So, the transformed function becomes:
[tex]\[
g(x) = \frac{1}{x - 2} + 7
\][/tex]
Putting it together, the function [tex]\( g(x) \)[/tex] after shifting 7 units up and 2 units to the right is:
[tex]\[
g(x) = \frac{1}{x - 2} + 7
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
