To determine the function \( g(x) \) from the parent function \( f(x) = \frac{1}{x} \) after shifting 4 units up and 3 units to the left, we need to apply the appropriate transformations to the function.
### Step-by-Step Solution:
1. Shifting the graph 3 units to the left:
- When we shift a function horizontally by \( h \) units to the left, we replace \( x \) with \( x + h \) in the equation of the function.
- Here, \( h = 3 \), so we replace \( x \) with \( x + 3 \) in the parent function \( f(x) = \frac{1}{x} \).
[tex]\[
f(x) \Rightarrow f(x + 3) = \frac{1}{x + 3}
\][/tex]
2. Shifting the graph 4 units up:
- When we shift a function vertically by \( k \) units up, we add \( k \) to the function.
- Here, \( k = 4 \), so we add 4 to the transformed function from the previous step.
[tex]\[
\frac{1}{x + 3} \Rightarrow \frac{1}{x + 3} + 4
\][/tex]
Thus, after shifting \( \frac{1}{x} \) 3 units to the left and 4 units up, the resulting function \( g(x) \) is:
[tex]\[
g(x) = \frac{1}{x + 3} + 4
\][/tex]
### Conclusion:
The function \( g(x) \) can be constructed by applying the specified transformations to the parent function \( f(x) \). Comparing this to the options given:
- A. \( g(x) = \frac{1}{x-3} + 4 \)
- B. \( g(x) = \frac{1}{x+3} + 4 \)
- C. \( g(x) = \frac{1}{x-4} + 3 \)
- D. \( g(x) = \frac{1}{x+4} + 3 \)
The correct function is given by option:
[tex]\[
\boxed{B}
\][/tex]
