To find the vertical asymptotes of the rational function [tex]\( g(x) = \frac{x-3}{x(x+5)} \)[/tex], we need to determine where the denominator of the function is equal to zero, as these are the points where the function is undefined and potentially has vertical asymptotes.
The denominator of the function is [tex]\( x(x+5) \)[/tex]. We need to solve the equation:
[tex]\[ x(x + 5) = 0. \][/tex]
Solving for [tex]\( x \)[/tex]:
1. Set each factor in the denominator equal to zero:
[tex]\[ x = 0 \][/tex]
[tex]\[ x + 5 = 0 \][/tex]
2. Solving these equations, we get:
[tex]\[ x = 0 \][/tex]
[tex]\[ x = -5 \][/tex]
These values, [tex]\( x = 0 \)[/tex] and [tex]\( x = -5 \)[/tex], are where the function [tex]\( g(x) \)[/tex] is undefined, indicating the locations of the vertical asymptotes.
Thus, the vertical asymptotes of the function [tex]\( g(x) = \frac{x-3}{x(x+5)} \)[/tex] are at:
[tex]\[ x = 0 \quad \text{and} \quad x = -5. \][/tex]
Therefore, the correct answer is:
B. [tex]\( x = 0 \)[/tex] and [tex]\( x = -5 \)[/tex]
