To determine how the graph of [tex]\( g(x) = \frac{1}{x-5} + 2 \)[/tex] compares to the parent function [tex]\( f(x) = \frac{1}{x} \)[/tex], we will look at the transformations applied to [tex]\( f(x) \)[/tex].
1. Horizontal Shift:
- The term [tex]\( x-5 \)[/tex] inside the function [tex]\( \frac{1}{x-5} \)[/tex] indicates a horizontal shift.
- Specifically, subtracting 5 from [tex]\( x \)[/tex] results in a horizontal shift to the right by 5 units.
2. Vertical Shift:
- The term [tex]\( +2 \)[/tex] outside the function [tex]\(\frac{1}{x-5}\)[/tex] indicates a vertical shift.
- Adding 2 results in a vertical shift upwards by 2 units.
Putting these transformations together:
- The function [tex]\( f(x) = \frac{1}{x} \)[/tex] is first shifted 5 units to the right to get [tex]\( \frac{1}{x-5} \)[/tex].
- Then, it is shifted 2 units up to reach [tex]\( \frac{1}{x-5} + 2 \)[/tex].
Therefore, the correct answer is:
[tex]\[ g(x) \text{ is shifted 5 units right and 2 units up from } f(x). \][/tex]
