To find the function [tex]\( g(x) \)[/tex] after shifting the graph of [tex]\( f(x) = \frac{1}{x} \)[/tex] 3 units up and 4 units to the right, we need to understand how these transformations affect the function.
### Horizontal Shift
A horizontal shift to the right involves replacing [tex]\( x \)[/tex] with [tex]\( x - h \)[/tex] in the function, where [tex]\( h \)[/tex] is the number of units shifted. For a shift 4 units to the right:
For [tex]\( f(x) = \frac{1}{x} \)[/tex], after shifting 4 units to the right, we get:
[tex]\[ f(x - 4) = \frac{1}{x - 4} \][/tex]
### Vertical Shift
A vertical shift up involves adding a constant [tex]\( k \)[/tex] to the function, where [tex]\( k \)[/tex] is the number of units shifted. For a shift 3 units up:
For [tex]\( \frac{1}{x - 4} \)[/tex], after shifting 3 units up, we add 3 to the function:
[tex]\[ g(x) = \frac{1}{x - 4} + 3 \][/tex]
### Conclusion Regarding the Choices
Let's check the answers provided:
A. [tex]\( g(x) = \frac{1}{x - 3} + 4 \)[/tex]
- This represents a shift 3 units to the right and 4 units up, which does not match the required shift.
B. [tex]\( g(x) = \frac{1}{x - 4} + 3 \)[/tex]
- This represents a shift 4 units to the right and 3 units up, which exactly matches the required shift.
C. [tex]\( g(x) = \frac{1}{x + 3} + 4 \)[/tex]
- This represents a shift 3 units to the left and 4 units up, which does not match the required shift.
D. [tex]\( g(x) = \frac{1}{x + 4} + 3 \)[/tex]
- This represents a shift 4 units to the left and 3 units up, which does not match the required shift.
Thus, the correct function [tex]\( g(x) \)[/tex] given the described transformations is:
[tex]\[ \boxed{g(x) = \frac{1}{x - 4} + 3} \][/tex]
