To transform the reciprocal parent function [tex]\( F(x) = \frac{1}{x} \)[/tex] according to the given shifts, we need to perform the following steps:
1. Shift 2 units left:
- Shifting the function to the left means adjusting the variable [tex]\(x\)[/tex] within the function. Specifically, to shift the function 2 units to the left, we replace [tex]\(x\)[/tex] with [tex]\((x + 2)\)[/tex].
Therefore, the function becomes:
[tex]\[
F(x + 2) = \frac{1}{x + 2}
\][/tex]
2. Shift 9 units up:
- Shifting the function up means adding a constant to the entire function. To shift the function up by 9 units, we simply add 9 to our already shifted function.
Thus, the new function becomes:
[tex]\[
G(x) = \frac{1}{x + 2} + 9
\][/tex]
Given these steps, the correct equation for the new function is [tex]\( G(x) = \frac{1}{x + 2} + 9 \)[/tex].
Therefore, the correct answer is:
C. [tex]\( G(x) = \frac{1}{x + 2} + 9 \)[/tex]
