At which values of [tex]x[/tex] does the function [tex]F(x)[/tex] have a vertical asymptote? Check all that apply.

[tex]F(x)=\frac{1}{(x-4)(x+1)}[/tex]

A. -1
B. -4
C. 4
D. 2
E. 0
F. 1



Answer :

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.