To determine the end behavior of the function [tex]\( f(x) = -2 \sqrt[3]{x} \)[/tex], we need to analyze what happens to [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] (positive infinity) and as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex] (negative infinity).
1. As [tex]\( x \rightarrow \infty \)[/tex]:
- Consider the cube root function [tex]\(\sqrt[3]{x}\)[/tex] when [tex]\( x \)[/tex] approaches [tex]\(\infty\)[/tex]:
- The cube root of a very large positive number is also a very large positive number.
- Therefore, [tex]\(\sqrt[3]{x} \rightarrow \infty\)[/tex] as [tex]\( x \rightarrow \infty \)[/tex].
- Now, [tex]\( f(x) = -2 \sqrt[3]{x} \)[/tex]:
- Since we multiply the cube root of a large positive number by -2, the resulting value will be a large negative number.
- Thus, as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
2. As [tex]\( x \rightarrow -\infty \)[/tex]:
- Consider the cube root function [tex]\(\sqrt[3]{x}\)[/tex] when [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex]:
- The cube root of a very large negative number is also a very large negative number.
- Therefore, [tex]\(\sqrt[3]{x} \rightarrow -\infty\)[/tex] as [tex]\( x \rightarrow -\infty \)[/tex].
- Now, [tex]\( f(x) = -2 \sqrt[3]{x} \)[/tex]:
- Since we multiply the cube root of a large negative number by -2, the resulting value will be a large positive number.
- Thus, as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
Putting it all together, the end behavior of the function [tex]\( f(x) = -2 \sqrt[3]{x} \)[/tex] is:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
Therefore, the correct end behavior is:
As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex], and as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
