Question

find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of the rational function.\\( h(x) = \frac{x + 3}{x(x + 1)} \\)\\( \dots \\)select the correct choice below and, if necessary, fill in the answer box to complete your choice. (type an equation. use a comma to separate answers as needed.)\\( \bigcirc \\) a. the vertical asymptote(s) is(are) \\( \square \\) and hole(s) corresponding to \\( \square \\).\\( \bigcirc \\) b. the vertical asymptote(s) is(are) \\( \square \\). there are no holes.\\( \bigcirc \\) c. there are no vertical asymptotes but there is(are) hole(s) corresponding to \\( \square \\).\\( \bigcirc \\) d. there are no discontinuities.

Explanation:

Step1: Factor Numerator and Denominator

The function is \( h(x)=\frac{x + 3}{x(x + 1)} \). The numerator \( x + 3 \) cannot be factored further, and the denominator is already factored as \( x(x + 1) \).

Step2: Find Values for Holes

Holes occur when a factor is common to both numerator and denominator. Here, numerator \( x + 3 \) and denominator \( x(x + 1) \) have no common factors. So, there are no holes.

Step3: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero (and numerator is not zero, since no holes). Set denominator \( x(x + 1)=0 \). Solving \( x = 0 \) or \( x+1 = 0\) (i.e., \( x=-1 \)). For \( x = 0 \), numerator is \( 0 + 3=3
eq0 \). For \( x=-1 \), numerator is \( -1 + 3 = 2
eq0 \). So vertical asymptotes are \( x = 0 \) and \( x=-1 \).

Answer:

B. The vertical asymptote(s) is(are) \( x = 0, x=-1 \). There are no holes.