The graph of the reciprocal parent function, [tex]f(x)=\frac{1}{x}[/tex], is shifted 4 units up and 3 units to the left to create the graph of [tex]g(x)[/tex]. What function is [tex]g(x)[/tex]?

A. [tex]g(x)=\frac{1}{x-3}+4[/tex]

B. [tex]g(x)=\frac{1}{x+3}+4[/tex]

C. [tex]g(x)=\frac{1}{x-4}+3[/tex]

D. [tex]g(x)=\frac{1}{x+4}+3[/tex]



Answer :

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]