https://file.aiquickdraw.com/mathbot/file/17255747595123v7qc62g.jpg

The problem asks which option describes the end behavior of a polynomial function, using limit notation as \( x \to \infty \) and \( x \to -\infty \).

Let's analyze each option one by one, based on the typical behavior of polynomial functions.

### Option 1:
\[
\lim_{x \to \infty} f(x) = -\infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = \infty
\]
This describes a polynomial with a negative leading coefficient and odd degree. For such functions:
- As \( x \to \infty \), the polynomial tends toward \( -\infty \).
- As \( x \to -\infty \), the polynomial tends toward \( \infty \).

### Option 2:
\[
\lim_{x \to \infty} f(x) = 8 \quad \text{and} \quad \lim_{x \to -\infty} f(x) = -8
\]
This behavior is inconsistent with a polynomial function. Polynomials grow without bound as \( x \to \infty \) or \( x \to -\infty \), so having limits that approach constants (like 8 and -8) suggests this isn't the correct option.

### Option 3:
\[
\lim_{x \to \pm \infty} f(x) = 1
\]
This suggests the function tends toward a constant as \( x \) approaches both positive and negative infinity, which is not possible for a polynomial function.

### Option 4:
\[
\lim_{x \to \infty} f(x) = \infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = -\infty
\]
This describes a polynomial with a positive leading coefficient and odd degree. For such functions:
- As \( x \to \infty \), the polynomial tends toward \( \infty \).
- As \( x \to -\infty \), the polynomial tends toward \( -\infty \).

### Conclusion:
The correct answer is **Option 4**, which describes the behavior of a polynomial with a positive leading coefficient and odd degree.

Do you have any questions or would you like more details?

---

Here are some related questions to expand on this topic:
1. What determines the end behavior of a polynomial function?
2. How can you determine the degree of a polynomial from its end behavior?
3. What is the role of the leading coefficient in determining the end behavior of polynomials?
4. How does the end behavior of even-degree polynomials differ from odd-degree polynomials?
5. How do polynomial functions differ from other functions, such as exponential functions, in terms of end behavior?

**Tip**: Always check the leading term of a polynomial to determine its end behavior, since the highest degree term dominates as \( x \to \infty \) or \( x \to -\infty \).

Which describes the end behavior of the polynomial function?

Analyze and determine the end behavior of polynomial functions using limits. This problem covers important concepts like the effect of the leading coefficient and degree of the polynomial on its end behavior, suitable for high school 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