https://file.aiquickdraw.com/mathbot/file/1726244269004jbovqv7k.jpg

The rational function provided in the image is:

\[
f(x) = \frac{-9(x - 5)(x^3 + 7)}{(x + 2)(x - 8)}
\]

To determine the end behavior of this rational function as \(x\) approaches positive infinity and negative infinity, we focus on the highest degree terms in both the numerator and denominator. 

### Step 1: Simplify the highest degree terms
- The highest degree term in the numerator is from \(x^3\) in \((x^3 + 7)\). When multiplied by \((x - 5)\), the highest degree term in the numerator is:
  \[
  -9x^4
  \]
- The highest degree term in the denominator is from the product \((x + 2)(x - 8)\), which simplifies to:
  \[
  x^2
  \]

Thus, the function \(f(x)\) behaves like:
\[
f(x) \approx \frac{-9x^4}{x^2} = -9x^2
\]

### Step 2: Analyze the end behavior
- As \(x \to +\infty\):
  \[
  f(x) \approx -9x^2 \to -\infty
  \]
- As \(x \to -\infty\):
  \[
  f(x) \approx -9x^2 \to -\infty
  \]

Thus, in both cases, the function approaches negative infinity as \(x\) approaches either positive or negative infinity.

### Conclusion:
The correct option is:
- As \(x\) approaches negative infinity, the values of \(f(x)\) approach negative infinity, and as \(x\) approaches positive infinity, the values of \(f(x)\) approach negative infinity.

This corresponds to the **third option**.

---

Would you like more details or further clarification?

Here are five related questions for additional practice:
1. How do you find horizontal asymptotes of rational functions?
2. How does the degree of a polynomial in the numerator and denominator affect the end behavior?
3. What are vertical asymptotes, and how can you find them from the given function?
4. What happens to \(f(x)\) if the highest degree in the numerator equals that in the denominator?
5. How does a negative leading coefficient affect the function's graph?

**Tip**: When determining end behavior, focus on the highest degree terms, as they dominate the behavior for large values of \(x\).

Given the rational function f(x) = -9(x - 5)(x^3 + 7) / ((x + 2)(x - 8)), which of the following is a true statement about the end behavior of f(x)?

Explore the end behavior of the rational function f(x) = -9(x - 5)(x^3 + 7) / ((x + 2)(x - 8)). Learn how the highest degree terms in the numerator and denominator affect the asymptotic behavior of the function as x approaches positive and negative infinity. This problem is suitable for advanced algebra and precalculus students in grades 10-12.

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