To determine the function [tex]\( g(x) \)[/tex] after shifting the reciprocal parent function [tex]\( f(x) = \frac{1}{x} \)[/tex] 7 units up and 2 units to the right, we'll follow these steps:
1. Shift 7 units up:
- When a function is shifted vertically upwards by [tex]\( k \)[/tex] units, we add [tex]\( k \)[/tex] to the function.
- Here, [tex]\( k = 7 \)[/tex], so the new function after shifting [tex]\( f(x) = \frac{1}{x} \)[/tex] up by 7 units is:
[tex]\[
f(x) + 7 = \frac{1}{x} + 7
\][/tex]
2. Shift 2 units to the right:
- When a function is shifted horizontally to the right by [tex]\( h \)[/tex] units, we replace [tex]\( x \)[/tex] with [tex]\( x - h \)[/tex].
- Here, [tex]\( h = 2 \)[/tex], so the new function after shifting [tex]\( f(x) = \frac{1}{x} \)[/tex] to the right by 2 units is:
[tex]\[
f(x - 2) = \frac{1}{x - 2}
\][/tex]
- Now, incorporate this horizontal shift into the previously vertically shifted function:
[tex]\[
g(x) = \frac{1}{x - 2} + 7
\][/tex]
Therefore, the function [tex]\( g(x) \)[/tex] after shifting the graph of [tex]\( f(x) = \frac{1}{x} \)[/tex] 7 units up and 2 units to the right is:
[tex]\[
g(x) = \frac{1}{x - 2} + 7
\][/tex]
Thus, the correct answer is:
A. [tex]\( g(x) = \frac{1}{x-2} + 7 \)[/tex]
