To find the function [tex]\( g(x) \)[/tex] after shifting the graph of the reciprocal parent function [tex]\( f(x) = \frac{1}{x} \)[/tex] 6 units down and 1 unit to the right, let's proceed step-by-step:
1. Shift 1 unit to the right:
- Shifting a function [tex]\( f(x) \)[/tex] to the right by [tex]\( a \)[/tex] units means replacing [tex]\( x \)[/tex] with [tex]\( x - a \)[/tex].
- Here, we need to shift [tex]\( f(x) = \frac{1}{x} \)[/tex] to the right by 1 unit. Hence, replace [tex]\( x \)[/tex] with [tex]\( x - 1 \)[/tex]:
[tex]\[
f(x) = \frac{1}{x} \implies f(x - 1) = \frac{1}{x - 1}
\][/tex]
Now, the function is [tex]\( f(x) = \frac{1}{x-1} \)[/tex].
2. Shift 6 units down:
- Shifting a function [tex]\( f(x) \)[/tex] down by [tex]\( b \)[/tex] units means subtracting [tex]\( b \)[/tex] from the function.
- Here, we need to shift [tex]\( f(x) = \frac{1}{x-1} \)[/tex] 6 units downwards. Therefore, we subtract 6 from the function:
[tex]\[
f(x) = \frac{1}{x-1} \implies f(x) - 6 = \frac{1}{x-1} - 6
\][/tex]
Now the function is [tex]\( g(x) = \frac{1}{x-1} - 6 \)[/tex].
Therefore, after shifting the graph of the reciprocal parent function [tex]\( f(x) = \frac{1}{x} \)[/tex] 6 units down and 1 unit to the right, the resulting function [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = \frac{1}{x-1} - 6
\][/tex]
So the correct answer is:
[tex]\[
\boxed{g(x)=\frac{1}{x-1}-6}
\][/tex]
