Question: The function f(x) is defined below. What is the end behavior of f(x)?
f, of, x, equals, 120, x, to the power 4 , minus, 2112, x, plus, 44, x, to the power 5 , minus, 1152, plus, 4, x, to the power 6 , minus, 1280, x, squared, minus, 160, x, cubed
f(x)=120x^4 - 2112x + 44x^5 - 1152 + 4x^6 - 1280x^2 - 160x^3
Answer
Attempt 1 out of 2
The number of attempts on this specific problem before you are marked incorrect
Multiple Choice Answers
as , x, right arrow, infinity, comma, f, of, x, right arrow, infinity, and , . , as , x, right arrow, minus, infinity, comma, f, of, x, right arrow, minus, infinity
as x\to\infty,f(x)\to\infty and as x\to-\infty,f(x)\to-\infty
as , x, right arrow, infinity, comma, f, of, x, right arrow, infinity, and , . , as , x, right arrow, minus, infinity, comma, f, of, x, right arrow, infinity
as x\to\infty,f(x)\to\infty and as x\to-\infty,f(x)\to\infty
as , x, right arrow, infinity, comma, f, of, x, right arrow, minus, infinity, and , . , as , x, right arrow, minus, infinity, comma, f, of, x, right arrow, infinity
as x\to\infty,f(x)\to-\infty and as x\to-\infty,f(x)\to\infty
as , x, right arrow, infinity, comma, f, of, x, right arrow, minus, infinity, and , . , as , x, right arrow, minus, infinity, comma, f, of, x, right arrow, minus, infinity
as x\to\infty,f(x)\to-\infty and as x\to-\infty,f(x)\to-\infty
1. **State the problem:** Determine the end behavior of the function $$f(x) = 120x^4 - 2112x + 44x^5 - 1152 + 4x^6 - 1280x^2 - 160x^3$$.
2. **Identify the leading term:** The end behavior of a polynomial function is dominated by the term with the highest power of $$x$$. Here, the highest power is $$6$$, so the leading term is $$4x^6$$.
3. **Recall the rule for end behavior:**
- If the leading coefficient is positive and the degree is even, then as $$x \to \infty$$, $$f(x) \to \infty$$ and as $$x \to -\infty$$, $$f(x) \to \infty$$.
- If the leading coefficient is negative and the degree is even, then as $$x \to \infty$$, $$f(x) \to -\infty$$ and as $$x \to -\infty$$, $$f(x) \to -\infty$$.
4. **Apply the rule:** The leading coefficient is $$4$$, which is positive, and the degree $$6$$ is even.
5. **Conclusion:**
- As $$x \to \infty$$, $$f(x) \to \infty$$.
- As $$x \to -\infty$$, $$f(x) \to \infty$$.
Therefore, the end behavior of $$f(x)$$ is:
$$\boxed{\text{As } x \to \infty, f(x) \to \infty \text{ and as } x \to -\infty, f(x) \to \infty}$$.