Answer :
To determine the values of [tex]\( x \)[/tex] at which the function [tex]\( F(x) = \frac{9}{(x-1)(x-8)} \)[/tex] has vertical asymptotes, we'll follow these steps:
1. Identify the denominator of the function: The denominator is [tex]\((x-1)(x-8)\)[/tex].
2. Set the denominator equal to zero: The function will have vertical asymptotes at the values of [tex]\( x \)[/tex] that make the denominator zero. So, we solve:
[tex]\[ (x-1)(x-8) = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]: Set each factor in the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \][/tex]
[tex]\[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \][/tex]
Therefore, the function [tex]\( F(x) \)[/tex] has vertical asymptotes at [tex]\( x = 1 \)[/tex] and [tex]\( x = 8 \)[/tex].
4. Select the correct options: Based on the choices given:
- A. 1
- B. -9
- C. 8
- D. -1
- E. -8
- F. 9
The values where the function has vertical asymptotes are:
- A. 1
- C. 8
Thus, the answer is:
At [tex]\( x = 1 \)[/tex] and [tex]\( x = 8 \)[/tex].
1. Identify the denominator of the function: The denominator is [tex]\((x-1)(x-8)\)[/tex].
2. Set the denominator equal to zero: The function will have vertical asymptotes at the values of [tex]\( x \)[/tex] that make the denominator zero. So, we solve:
[tex]\[ (x-1)(x-8) = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]: Set each factor in the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \][/tex]
[tex]\[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \][/tex]
Therefore, the function [tex]\( F(x) \)[/tex] has vertical asymptotes at [tex]\( x = 1 \)[/tex] and [tex]\( x = 8 \)[/tex].
4. Select the correct options: Based on the choices given:
- A. 1
- B. -9
- C. 8
- D. -1
- E. -8
- F. 9
The values where the function has vertical asymptotes are:
- A. 1
- C. 8
Thus, the answer is:
At [tex]\( x = 1 \)[/tex] and [tex]\( x = 8 \)[/tex].
