To determine the number of vertical asymptotes for the function [tex]\( F(x)=\frac{3}{(x-11)(x+4)} \)[/tex], we need to see where the function is undefined. Vertical asymptotes occur when the denominator of the fraction is equal to zero and the numerator is not zero.
Given the function [tex]\( F(x) = \frac{3}{(x-11)(x+4)} \)[/tex], let's analyze the denominator:
1. Set the denominator [tex]\((x-11)(x+4) = 0\)[/tex] and solve for [tex]\( x \)[/tex].
2. The equation [tex]\((x-11)(x+4) = 0\)[/tex] can be solved by setting each factor equal to zero:
[tex]\( x - 11 = 0 \)[/tex] gives [tex]\( x = 11 \)[/tex]
[tex]\( x + 4 = 0 \)[/tex] gives [tex]\( x = -4 \)[/tex]
3. These solutions indicate the values of [tex]\( x \)[/tex] where the denominator is zero and thus where the function [tex]\( F(x) \)[/tex] is undefined. At these points, the function has vertical asymptotes.
Therefore, the graph of the function [tex]\( F(x) = \frac{3}{(x-11)(x+4)} \)[/tex] has vertical asymptotes at [tex]\( x = 11 \)[/tex] and [tex]\( x = -4 \)[/tex].
To summarize, there are 2 vertical asymptotes for the given function.
So, the correct answer is:
D. 2
