To determine the function [tex]\( g(x) \)[/tex] based on the transformations applied to the parent function [tex]\( f(x) = \frac{1}{x} \)[/tex], let's analyze the shifts step by step.
1. Right Shift by 1 Unit:
Shifting a function horizontally to the right by 1 unit means replacing [tex]\( x \)[/tex] with [tex]\( x - 1 \)[/tex] in the function [tex]\( f(x) \)[/tex].
Thus, the function [tex]\( f(x) = \frac{1}{x} \)[/tex] transforms to [tex]\( f(x) = \frac{1}{x - 1} \)[/tex].
2. Down Shift by 6 Units:
Shifting a function vertically down by 6 units means subtracting 6 from the function.
Therefore, the function [tex]\( \frac{1}{x - 1} \)[/tex] becomes [tex]\( g(x) = \frac{1}{x - 1} - 6 \)[/tex].
After applying both transformations, the resulting function is:
[tex]\[ g(x) = \frac{1}{x - 1} - 6 \][/tex]
Let's verify which option this matches:
A. [tex]\( g(x) = \frac{1}{x + 1} - 6 \)[/tex]
B. [tex]\( g(x) = \frac{1}{x - 6} - 1 \)[/tex]
C. [tex]\( g(x) = \frac{1}{x - 1} - 6 \)[/tex]
D. [tex]\( g(x) = \frac{1}{x - 6} - 1 \)[/tex]
The correct function [tex]\( g(x) \)[/tex] is indeed [tex]\( g(x) = \frac{1}{x - 1} - 6 \)[/tex], which corresponds to option C.
So the correct answer is:
[tex]\[ \boxed{3} \][/tex]
