To determine the values of [tex]\( x \)[/tex] at which the function [tex]\( F(x) \)[/tex] has vertical asymptotes, we need to identify when the denominator of the function equals zero, since division by zero is undefined and leads to vertical asymptotes.
The given function is:
[tex]\[ F(x) = \frac{1}{(x-4)(x+1)} \][/tex]
The denominator of this function is:
[tex]\[ (x-4)(x+1) \][/tex]
To find the vertical asymptotes, we set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ (x-4)(x+1) = 0 \][/tex]
This equation will be satisfied if any of the factors in the denominator are equal to zero. Therefore, we need to solve each factor separately:
1. [tex]\( x - 4 = 0 \)[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 4 \][/tex]
2. [tex]\( x + 1 = 0 \)[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = -1 \][/tex]
So, the values of [tex]\( x \)[/tex] for which the function [tex]\( F(x) \)[/tex] has vertical asymptotes are [tex]\( x = 4 \)[/tex] and [tex]\( x = -1 \)[/tex].
Thus, the correct answers are:
- A. -1
- C. 4
These are the values where the function [tex]\( F(x) \)[/tex] has vertical asymptotes.
