To analyze question 4, we determine the end behavior of the polynomial $f(x) = x^9 + 10x^4 - 4x^2 - 3$. This process involves identifying the highest-degree term and how it influences the function as $x \to +\infty$ and $x \to -\infty$.

Steps to Determine End Behavior:

1. Identify the leading term: The term with the highest degree in the polynomial is $x^9$. Since the coefficient of $x^9$ is positive ($+1$), this term determines the end behavior.

2. Analyze as $x \to +\infty$: When $x \to +\infty$, the term $x^9$ dominates, and since $x^9 > 0$, $f(x) \to +\infty$.

3. Analyze as $x \to -\infty$: When $x \to -\infty$, $x^9$ (an odd-degree term) becomes negative, so $f(x) \to -\infty$.

4. Proper Notation:

   • As $x \to +\infty$, $f(x) \to +\infty$.
   • As $x \to -\infty$, $f(x) \to -\infty$.

Final Answer:

The end behavior of $f(x) = x^9 + 10x^4 - 4x^2 - 3$ is: $\text{As } x \to +\infty, f(x) \to +\infty. \quad \text{As } x \to -\infty, f(x) \to -\infty.$

Would you like me to break down any specific step or verify another part of the question?

Relative Questions:

1. How do odd-degree polynomials differ from even-degree polynomials in terms of end behavior?
2. What is the role of the coefficient of the leading term in determining end behavior?
3. Why does the $x^9$ term dominate over other terms as $x \to \pm\infty$?
4. How would the end behavior change if the leading term had a negative coefficient?
5. Can we apply the same method to rational functions?

Tip: Always focus on the leading term to determine the end behavior, as it overpowers all other terms for large $|x|$.