To analyze the end behavior of the function \( f(x) = \frac{8x^2 + 3x + 17}{x^2 + 4} \), we focus on the behavior of the function as \( x \) approaches infinity (\( x \to \infty \)) and negative infinity (\( x \to -\infty \)).

### Step 1: Identify the degree of the polynomial in the numerator and denominator.
- **Numerator:** \( 8x^2 + 3x + 17 \) has a degree of 2.
- **Denominator:** \( x^2 + 4 \) also has a degree of 2.

### Step 2: Determine the leading terms.
For large values of \( x \), the leading term in both the numerator and denominator will dominate the behavior of the function. Thus, we approximate the function by focusing on these leading terms:
\[
f(x) \approx \frac{8x^2}{x^2} = 8
\]

### Step 3: Evaluate the end behavior.
- As \( x \to \infty \), \( f(x) \approx 8 \).
- As \( x \to -\infty \), \( f(x) \approx 8 \).

### Conclusion:
The end behavior of the function \( f(x) \) is such that as \( x \) approaches either positive or negative infinity, the function approaches the horizontal asymptote \( y = 8 \).

Would you like more details on this explanation or have any further questions?

---

Here are five related questions to consider:
1. What are the horizontal asymptotes of the function \( g(x) = \frac{5x^3 + 2x + 1}{2x^3 + 3x^2 + 4} \)?
2. How do you determine vertical asymptotes for a rational function?
3. What is the significance of the leading term in polynomial functions for end behavior?
4. How does the end behavior of \( f(x) \) change if the degree of the numerator is higher than the degree of the denominator?
5. Can the function \( f(x) \) have any slant (oblique) asymptotes?

**Tip:** When analyzing the end behavior of a rational function, focus on the leading terms of the numerator and denominator. This often simplifies the process.

Understand the end behavior of the function f(x) = (8x^2 + 3x + 17) / (x^2 + 4) as x approaches infinity and negative infinity. Learn about rational functions, asymptotes, and how to analyze end behavior in algebra. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation