Determine the end behavior of the function as x → +∞ and as x → −∞.
f(x) = 1000 − 39x + 51x^2 − 10x^3
As x → +∞, f(x) → ????
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As x → −∞, f(x) → ???
I know that the end behavior of the function is determined by the leading coefficient which in this case would be 10x^3? I just don't know what values I should plug into the answer box. Will the end function approaching both negative infinity and positive infinity just be positive?
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the "dominating" term is -10x^3 , not 10x^3
so as x → +∞ , -10x^3 → -∞ , and
as x→ -∞ , -10x^3 → +∞
general appearance ...
http://www.wolframalpha.com/input/?i=f%28x%29+%3D+1000+%E2%88%92+39x+%2B+51x%5E2+%E2%88%92+10x%5E3+
Thank you so much! I missed the negative part about the term which definitely impacted the answer.
To determine the end behavior of the function f(x) = 1000 - 39x + 51x^2 - 10x^3 as x approaches +∞ and -∞, you are correct that we look at the leading term, which is -10x^3.
1. As x approaches +∞:
When x approaches positive infinity, the term -10x^3 becomes very large and negative. The other terms (1000, -39x, and 51x^2) become insignificant compared to -10x^3. Therefore, as x approaches +∞, f(x) approaches negative infinity (or -∞).
2. As x approaches -∞:
When x approaches negative infinity, the term -10x^3 becomes very large and negative. The other terms (1000, -39x, and 51x^2) become insignificant. Therefore, as x approaches -∞, f(x) also approaches negative infinity (or -∞).
In conclusion:
As x approaches +∞, f(x) approaches -∞.
As x approaches -∞, f(x) also approaches -∞.
To determine the end behavior of the function f(x) = 1000 − 39x + 51x^2 − 10x^3, we need to look at the highest power term in the polynomial, which is the term with the leading coefficient.
In this case, the highest power term is -10x^3. The sign of the leading coefficient (-10) tells us how the function behaves as x approaches positive or negative infinity.
1. As x approaches positive infinity (x → +∞), we need to consider the sign of the leading coefficient (-10). Since the leading coefficient is negative, the function will approach negative infinity. So, as x approaches positive infinity, f(x) → -∞.
2. As x approaches negative infinity (x → -∞), we again consider the sign of the leading coefficient (-10). Since the leading coefficient is negative, the function will approach negative infinity. So, as x approaches negative infinity, f(x) → -∞.
Therefore, both as x approaches positive infinity and as x approaches negative infinity, the function f(x) approaches negative infinity.
In summary:
- As x → +∞, f(x) → -∞ (negative infinity).
- As x → -∞, f(x) → -∞ (negative infinity).