At which value(s) of x does the graph of the function F(x) have a vertical asymptote? Check all that apply.

𝐹

(

𝑥

)

=

𝑥

(

𝑥

+

2

)

(

𝑥

−

1

)

F(x)= 

(x+2)(x−1)

x

​
 

A.

x = 1

B.

x = -2

C.

x = 10

D.

x = 0

E.

x = 2

F.

x = -1

SUBMIT

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To determine the value(s) of \( x \) where the function \( F(x) \) has a vertical asymptote, we need to examine the points where the denominator of the function equals zero. Vertical asymptotes occur where the denominator is zero, and the numerator is not zero at the same points.

### Function Given:
\[
F(x) = \frac{(x+2)(x-1)}{x}
\]

### Denominator:
The denominator is \( x \). The function will have a vertical asymptote wherever the denominator is zero. So, we set the denominator equal to zero:
\[
x = 0
\]

### Conclusion:
- The vertical asymptote occurs at **\( x = 0 \)**.

### Correct Answer:
- **D. \( x = 0 \)**

There are no other values of \( x \) that make the denominator zero, so only \( x = 0 \) is correct.

Would you like a more detailed explanation or have any other questions? Here are some related questions you might find useful:

1. How do you find horizontal asymptotes for a rational function?
2. What happens to \( F(x) \) as \( x \) approaches the vertical asymptote?
3. Can a function have more than one vertical asymptote?
4. What would the graph look like near the vertical asymptote at \( x = 0 \)?
5. How do you determine if there is a hole (removable discontinuity) in the graph?

**Tip:** Remember that vertical asymptotes occur where the function is undefined due to division by zero, but where the numerator isn't zero.

Learn how to identify vertical asymptotes in rational functions with the function F(x) = (x+2)(x-1)/x. Understand where the function is undefined due to division by zero and the conditions under which vertical asymptotes occur. Suitable for high school students.

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