The graph of the reciprocal parent function, [tex]f(x)=\frac{1}{x}[/tex], is shifted 5 units up and 8 units to the left to create the graph of [tex]g(x)[/tex]. What function is [tex]g(x)[/tex]?

A. [tex]g(x)=\frac{1}{x-5}+8[/tex]

B. [tex]g(x)=\frac{1}{x-8}+5[/tex]

C. [tex]g(x)=\frac{1}{x+5}+8[/tex]

D. [tex]g(x)=\frac{1}{x+8}+5[/tex]



Answer :

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]