To understand how the graph of [tex]\( g(x) = \frac{1}{x-5} + 2 \)[/tex] compares to the graph of the parent function [tex]\( f(x) = \frac{1}{x} \)[/tex], let's analyze the transformations involved.
1. Horizontal Shift:
- The term inside the function [tex]\( \frac{1}{x-5} \)[/tex] indicates a horizontal shift.
- For the function [tex]\( f(x) = \frac{1}{x} \)[/tex], when we replace [tex]\( x \)[/tex] with [tex]\( x - h \)[/tex], where [tex]\( h \)[/tex] is a constant, it represents a horizontal shift of [tex]\( h \)[/tex] units.
- Specifically, [tex]\( \frac{1}{x-5} \)[/tex] means that [tex]\( x \)[/tex] is replaced by [tex]\( x - 5 \)[/tex], which corresponds to a horizontal shift of 5 units to the right.
2. Vertical Shift:
- The term outside the function, [tex]\( +2 \)[/tex], indicates a vertical shift.
- For the function [tex]\( f(x) = \frac{1}{x} \)[/tex], when we add a constant [tex]\( k \)[/tex], it represents a vertical shift of [tex]\( k \)[/tex] units.
- In this case, adding 2 means the function is shifted 2 units up.
Therefore, the graph of [tex]\( g(x) = \frac{1}{x-5} + 2 \)[/tex] is the graph of [tex]\( f(x) = \frac{1}{x} \)[/tex] shifted 5 units to the right and 2 units up.
So, the correct comparison is:
[tex]\[ g(x) \text{ is shifted 5 units right and 2 units up from } f(x). \][/tex]
