https://file.aiquickdraw.com/mathbot/file/1730682377306b8x9neju.jpg

To solve this problem, let's analyze the given function:

\[
f(x) = \frac{x}{x + 8}
\]

### Step 1: Determine Vertical Asymptotes
Vertical asymptotes occur where the denominator of the function is zero, provided the numerator is not also zero at the same point (which would indicate a hole instead).

For \( f(x) = \frac{x}{x + 8} \), the denominator is \( x + 8 \). Set the denominator to zero:

\[
x + 8 = 0 \implies x = -8
\]

Since the numerator \( x \) does not also equal zero at \( x = -8 \), this value represents a vertical asymptote rather than a hole.

### Step 2: Check for Holes
Holes occur when both the numerator and denominator have a common factor that becomes zero. In this case, there are no factors in the numerator that would cancel with the denominator term \( x + 8 \), so there are no holes in the function.

### Conclusion
- **Vertical Asymptote**: \( x = -8 \)
- **No Holes**

### Correct Answer Choice
The correct answer is:
- **C.** The vertical asymptote(s) is(are) \( x = -8 \). There are no holes.

Would you like further details or have any questions?

---

### Related Questions
1. What would happen to the asymptotes if the function were \( f(x) = \frac{x+8}{x+8} \)?
2. How would we identify horizontal asymptotes for a rational function?
3. What defines the difference between a hole and a vertical asymptote?
4. How can we graphically represent holes and vertical asymptotes in rational functions?
5. What is the significance of factoring in finding holes in rational functions?

### Tip
Remember, a vertical asymptote happens when the denominator is zero without being canceled by the numerator, while a hole occurs if both the numerator and denominator share a common factor that zeroes out.

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of the rational function f(x) = x / (x + 8).

Explore how to find the vertical asymptotes and determine if there are any holes in the rational function f(x) = x / (x + 8). This example covers the steps to analyze the function's denominator and numerator for vertical asymptotes and holes, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation