To determine the function [tex]\( g(x) \)[/tex] that results from shifting the graph of the reciprocal parent function [tex]\( f(x) = \frac{1}{x} \)[/tex] both horizontally and vertically, we need to apply the respective transformations step by step.
1. Horizontal Shift:
The function [tex]\( f(x) \)[/tex] shifts horizontally to the right by 4 units. When you shift a function horizontally to the right by [tex]\( h \)[/tex] units, you replace [tex]\( x \)[/tex] with [tex]\( x - h \)[/tex]. For our function:
[tex]\[
f(x - 4) = \frac{1}{x - 4}
\][/tex]
2. Vertical Shift:
Next, the function undergoes a vertical shift upwards by 3 units. When you shift a function vertically up by [tex]\( k \)[/tex] units, you add [tex]\( k \)[/tex] to the entire function. Applying this to our horizontally shifted function:
[tex]\[
f(x - 4) + 3 = \frac{1}{x - 4} + 3
\][/tex]
Therefore, the final transformed function [tex]\( g(x) \)[/tex] after the described shifts is:
[tex]\[
g(x) = \frac{1}{x - 4} + 3
\][/tex]
By looking at the options provided:
A. [tex]\( g(x) = \frac{1}{x + 3} + 4 \)[/tex]
B. [tex]\( g(x) = \frac{1}{x - 3} + 4 \)[/tex]
C. [tex]\( g(x) = \frac{1}{x - 4} + 3 \)[/tex]
D. [tex]\( g(x) = \frac{1}{x + 4} + 3 \)[/tex]
We find that the correct option is:
[tex]\[
\boxed{C}
\][/tex]
