To find the function [tex]\( g(x) \)[/tex] after shifting the reciprocal parent function [tex]\( f(x) = \frac{1}{x} \)[/tex], we need to properly understand how horizontal and vertical shifts impact the function.
1. Horizontal Shift:
A shift to the left by 8 units means that for every [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex], we replace [tex]\( x \)[/tex] with [tex]\( x + 8 \)[/tex]. This is because substituting [tex]\( x \)[/tex] with [tex]\( x + 8 \)[/tex] in the function [tex]\( f(x) \)[/tex] essentially moves the graph to the left by 8 units:
[tex]\[
f(x + 8) = \frac{1}{x + 8}
\][/tex]
2. Vertical Shift:
A shift up by 5 units involves simply adding 5 to the function. This is because adding a constant to the function [tex]\( f(x) \)[/tex] shifts its graph vertically:
[tex]\[
f(x + 8) + 5 = \frac{1}{x + 8} + 5
\][/tex]
Combining these two transformations, we get the new function [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[ g(x) = \frac{1}{x + 8} + 5 \][/tex]
Therefore, the correct option is:
[tex]\[ \text{D. } g(x) = \frac{1}{x + 8} + 5 \][/tex]
