To determine the end behavior of the function [tex]\( f(x) = -4 \sqrt[3]{x} \)[/tex], we need to analyze how the function behaves as [tex]\( x \)[/tex] approaches negative infinity and positive infinity.
1. As [tex]\( x \rightarrow -\infty \)[/tex]:
- When [tex]\( x \)[/tex] is a large negative number, the cube root of [tex]\( x \)[/tex] (which is [tex]\( \sqrt[3]{x} \)[/tex]) will also be a negative number, but relatively smaller in magnitude since the cube root of a very large negative number is still negative but not as large as [tex]\( x \)[/tex] itself.
- Multiplying this negative value by [tex]\(-4\)[/tex] (a negative constant) will result in a large positive value.
Therefore, as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
2. As [tex]\( x \rightarrow \infty \)[/tex]:
- When [tex]\( x \)[/tex] is a large positive number, the cube root of [tex]\( x \)[/tex], [tex]\( \sqrt[3]{x} \)[/tex], will be a positive number.
- Multiplying this positive value by [tex]\(-4\)[/tex] will result in a large negative value.
Therefore, as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
Based on these analyses, the correct end behavior of the function is:
As [tex]\( x \rightarrow -\infty, f(x) \rightarrow \infty \)[/tex], and as [tex]\( x \rightarrow \infty, f(x) \rightarrow -\infty \)[/tex].
