Determine the vertical​ asymptote(s) of the following function. If none​ exists, state that fact.

f(x) = (x+3) / (x^2 + 11x + 24)


To determine the vertical asymptote(s) of the function 

\[
f(x) = \frac{x+3}{x^2 + 11x + 24},
\]

we first need to find the values of \( x \) that make the denominator equal to zero, as vertical asymptotes occur when the denominator is zero and the numerator is not zero at those points.

### Step 1: Factor the denominator
The denominator is \( x^2 + 11x + 24 \). We can factor this quadratic expression:

\[
x^2 + 11x + 24 = (x + 3)(x + 8).
\]

### Step 2: Set the denominator equal to zero
To find the vertical asymptotes, set the denominator equal to zero:

\[
(x + 3)(x + 8) = 0.
\]

This gives us two potential vertical asymptotes:

\[
x + 3 = 0 \quad \text{or} \quad x + 8 = 0,
\]
which simplifies to:

\[
x = -3 \quad \text{or} \quad x = -8.
\]

### Step 3: Check for cancellation
Now, check if the numerator \( x + 3 \) can cancel with the denominator. Since the numerator also contains \( x + 3 \), the factor \( x + 3 \) cancels out with the same factor in the denominator. Therefore, \( x = -3 \) is **not** a vertical asymptote; instead, it is a **removable discontinuity**.

### Conclusion:
The only vertical asymptote is at \( x = -8 \), because the factor \( x + 8 \) does not cancel out.

### Final Answer:
The function has a vertical asymptote at \( x = -8 \).

Would you like further explanation on this process, or do you have any other questions?

### Related Questions:
1. What are horizontal asymptotes and how do we find them for this function?
2. How do removable discontinuities differ from vertical asymptotes?
3. Can a function have both vertical and horizontal asymptotes? Why or why not?
4. How do you determine horizontal asymptotes for rational functions?
5. What happens graphically at a removable discontinuity?

### Tip:
Always check for common factors between the numerator and denominator to identify removable discontinuities before concluding the vertical asymptotes.

Determine the vertical asymptote(s) of the following function. If none exists, state that fact.

f(x) = (x+3) / (x^2 + 11x + 24)

Learn how to determine the vertical asymptotes of the rational function f(x) = (x+3) / (x^2 + 11x + 24). This step-by-step guide covers key concepts like factorization, vertical asymptotes, and removable discontinuities. Suitable for high school students in Grades 10-12.

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