how does the graph of $g(x)=\frac{1}{x - 5}+2$ compare to the graph of the parent function…

Explanation:

Step1: Analyze horizontal shift

For a function of the form $y = \frac{1}{x - h}+k$ compared to $y=\frac{1}{x}$, the horizontal shift is determined by $h$. In $g(x)=\frac{1}{x - 5}+2$, since $h = 5$, the graph of $y=\frac{1}{x}$ shifts 5 units to the right. (When $y=\frac{1}{x - h}$, if $h>0$, shift right; if $h < 0$, shift left).

Step2: Analyze vertical shift

The vertical shift is determined by $k$. In $g(x)=\frac{1}{x - 5}+2$, since $k = 2$, the graph of $y=\frac{1}{x}$ shifts 2 units up. (When $y=\frac{1}{x}+k$, if $k>0$, shift up; if $k < 0$, shift down).

Answer:

$g(x)$ is shifted 5 units right and 2 units up from $f(x)$.