f ( x )

The function given is:

$f(x) = -2(x - 1)^3(x + 2)^2$

To determine the end behavior, let's examine the leading term:

1. As $x \to \infty$ (large positive $x$):

   • The factor $(x - 1)^3$ behaves like $x^3$ for large $x$, and the factor $(x + 2)^2$ behaves like $x^2$.
   • So, the function behaves like: $f(x) \approx -2x^3 \cdot x^2 = -2x^5$
     • As $x \to \infty$, $-2x^5 \to -\infty$. Therefore: $\lim_{x \to \infty} f(x) = -\infty$

2. As $x \to -\infty$ (large negative $x$):

   • Again, $(x - 1)^3$ behaves like $x^3$ and $(x + 2)^2$ behaves like $x^2$.
   • The function behaves like: $f(x) \approx -2x^3 \cdot x^2 = -2x^5$
     • As $x \to -\infty$, $-2x^5 \to -\infty$. Therefore: $\lim_{x \to -\infty} f(x) = -\infty$

Conclusion:

The correct answer is:

$\text{as } x \to \infty, f(x) \to -\infty; \quad \text{as } x \to -\infty, f(x) \to -\infty$

Let me know if you'd like more details on any part of the solution!

Related Questions:

1. How do even and odd powers of terms affect the end behavior of a function?
2. What would happen if the exponent of $(x-1)$ was even instead of odd?
3. How do we determine the end behavior of a rational function (one with a numerator and denominator)?
4. How would the end behavior change if the coefficient of the function was positive instead of negative?
5. How can we graph the function to confirm the end behavior?

Tip: When determining the end behavior of polynomial functions, focus on the highest-degree terms, as they dominate for very large or very small values of $x$.