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What are the vertical asymptotes of the function above? 

Question 1Answer

a.
x = -3, x =-1


b.
y = 3, y = 1


c.
x = 1, x =-3


d.
x = 1, x =3 

The function you provided is:

\[
f(x) = \frac{x - 1}{x^2 + 4x + 3}
\]

### To find the vertical asymptotes:
Vertical asymptotes occur when the denominator equals zero, as long as the numerator does not also equal zero at the same points. We need to solve the equation:

\[
x^2 + 4x + 3 = 0
\]

#### Factor the quadratic:
\[
x^2 + 4x + 3 = (x + 3)(x + 1)
\]

Now, set each factor equal to zero:

\[
x + 3 = 0 \quad \text{or} \quad x + 1 = 0
\]

Solving these:

\[
x = -3 \quad \text{or} \quad x = -1
\]

Therefore, the vertical asymptotes are \(x = -3\) and \(x = -1\).

### Answer: 
The correct option is:

**a. \( x = -3 \), \( x = -1 \)**

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Do you want further details or have any questions?

### Related Questions:
1. How do we determine the horizontal asymptotes of this function?
2. What happens to the function as \(x \to \infty\)?
3. Can this function have any oblique asymptotes?
4. How does the behavior of the function near \(x = -3\) and \(x = -1\) compare?
5. What is the domain of the function?

### Tip:
Always check both the numerator and the denominator when finding asymptotes to ensure no cancellations occur.

What are the vertical asymptotes of the function f(x) = (x - 1) / (x^2 + 4x + 3)?

Learn how to find the vertical asymptotes of the function f(x) = (x - 1) / (x^2 + 4x + 3) by factoring the denominator and solving for x. This problem involves key concepts in algebra and rational functions, suitable for high school students in Grades 9-12.

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