The question asks which of the given functions $f(x)$ approach negative infinity as $x \to -\infty$.

To determine the behavior of each function as $x \to -\infty$, we need to focus on the term with the highest degree in each polynomial. This term will dominate the behavior as $x$ becomes very large (in the negative direction).

Let's analyze each function:

1. $f(x) = -5x^8$:

   • The highest degree term is $-5x^8$.
   • As $x \to -\infty$, $x^8 \to \infty$ since it's an even power, but the coefficient is negative, so $f(x) \to -\infty$.

2. $f(x) = 7x^4$:

   • The highest degree term is $7x^4$.
   • $x^4 \to \infty$ as $x \to -\infty$ because it is an even power. Since the coefficient is positive, $f(x) \to \infty$, not negative infinity.

3. $f(x) = x^3 + 18$:

   • The highest degree term is $x^3$.
   • As $x \to -\infty$, $x^3 \to -\infty$ since it's an odd power. Therefore, $f(x) \to -\infty$.

4. $f(x) = -2x^5 - 3x + 26$:

   • The highest degree term is $-2x^5$.
   • As $x \to -\infty$, $x^5 \to -\infty$ (odd power) and the coefficient is negative, so $f(x) \to \infty$, not negative infinity.

5. $f(x) = 8x^2 + 9x - 45$:

   • The highest degree term is $8x^2$.
   • As $x \to -\infty$, $x^2 \to \infty$ (even power) and the coefficient is positive, so $f(x) \to \infty$.

6. $f(x) = -5x^3 + 14x^2 - 8x + 76$:

   • The highest degree term is $-5x^3$.
   • As $x \to -\infty$, $x^3 \to -\infty$ (odd power) and the coefficient is negative, so $f(x) \to \infty$, not negative infinity.

Answers:

• $f(x) = -5x^8$
• $f(x) = x^3 + 18$

Would you like further clarification or more detailed steps?

Here are five related questions for practice:

1. What is the behavior of $f(x) = 3x^2 - 5x + 10$ as $x \to -\infty$?
2. How do odd-degree polynomials behave as $x \to \infty$ and $x \to -\infty$?
3. What happens to the function $f(x) = -4x^7$ as $x \to \infty$?
4. For the function $f(x) = 2x^6 + 3x^4$, what happens as $x \to -\infty$?
5. Can a function with an even-degree leading term ever approach negative infinity as $x \to -\infty$?

Tip: Always look at the term with the highest degree when analyzing the end behavior of a polynomial function.