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The image shows a math problem asking to "Determine the end behavior of \( f(x) = x^3 - 2x^2 + x - 4 \)." Below are four multiple-choice options for the end behavior of the polynomial function.

To determine the end behavior of a polynomial, we focus on the leading term, which is the term with the highest degree. In this case, the leading term is \( x^3 \).

For \( f(x) = x^3 - 2x^2 + x - 4 \):
- The degree of the polynomial is 3 (odd degree).
- The coefficient of the leading term is positive (1).

The general rule for polynomials is:
- For an odd-degree polynomial with a positive leading coefficient, as \( x \to +\infty \), \( f(x) \to +\infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).

Thus, the end behavior is:
- As \( x \to +\infty \), \( f(x) \to +\infty \)
- As \( x \to -\infty \), \( f(x) \to -\infty \)

Now, you can select the option that matches this behavior.

Would you like further details on how this is calculated or any other assistance? 

Here are 5 related questions to explore:
1. What are the end behaviors of even-degree polynomials with negative leading coefficients?
2. How do you find the degree and leading term of more complex polynomials?
3. Can the end behavior of a function be affected by non-leading terms?
4. What are some real-world applications of polynomial end behavior?
5. How can graphing a polynomial function help confirm the end behavior?

**Tip**: When analyzing polynomial functions, always start by identifying the leading term for insights on end behavior.

Determine the end behavior of f(x) = x^3 - 2x^2 + x - 4.

Understand how to determine the end behavior of the polynomial function f(x) = x^3 - 2x^2 + x - 4. This problem highlights key polynomial concepts such as the leading term and its influence on end behavior. Perfect for high school students in Grades 9-12.

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