Question
Describe the end behavior (long run behavior) of \( f(x)=-x^{8} \) As \( x \rightarrow-\infty, f(x) \rightarrow \) ? As \( x \rightarrow \infty, f(x) \rightarrow \) ? ?
Ask by Morgan Nguyen. in the United States
Mar 17,2025
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Answer
As \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity. As \( x \) approaches positive infinity, \( f(x) \) also approaches negative infinity.
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As \( x \rightarrow -\infty, f(x) \rightarrow +\infty \). This is because the leading term \( -x^8 \) is negative, and since it's raised to an even power, the output will positive when \( x \) is a large negative number. As \( x \rightarrow \infty, f(x) \rightarrow -\infty \). Here, when \( x \) takes on large positive values, the term \( -x^8 \) becomes vastly negative due to the negative coefficient and the even exponent, leading \( f(x) \) to trend downwards.