Certainly! Let's carefully determine the equation of the new function [tex]\( g(x) \)[/tex] based on the given transformations:
1. Starting Point:
The parent function is:
[tex]\[
f(x) = \frac{1}{x}
\][/tex]
2. Shifting 8 Units to the Left:
When you shift a function horizontally, you replace [tex]\( x \)[/tex] with [tex]\( x + c \)[/tex], where [tex]\( c \)[/tex] is the number of units shifted. For a shift 8 units to the left:
[tex]\[
f(x + 8) = \frac{1}{x + 8}
\][/tex]
3. Shifting 5 Units Up:
When you shift a function vertically, you add [tex]\( c \)[/tex] to the entire function, where [tex]\( c \)[/tex] is the number of units shifted up:
[tex]\[
g(x) = \frac{1}{x + 8} + 5
\][/tex]
So the function [tex]\( g(x) \)[/tex] after shifting the graph of the reciprocal function [tex]\( 5 \)[/tex] units up and [tex]\( 8 \)[/tex] units to the left is:
[tex]\[
g(x) = \frac{1}{x + 8} + 5
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{B. \ g(x) = \frac{1}{x + 8} + 5}
\][/tex]
