To determine the end behavior of the function [tex]\( f(x) = -2 \sqrt[3]{x} \)[/tex], let's analyze the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches positive and negative infinity.
1. As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \rightarrow \infty \)[/tex]):
- Consider a very large positive value of [tex]\( x \)[/tex], such as [tex]\( 10^9 \)[/tex].
- Substituting [tex]\( x = 10^9 \)[/tex] into the function:
[tex]\[
f(10^9) = -2 \left( 10^9 \right)^{1/3}
\][/tex]
- The cube root of [tex]\( 10^9 \)[/tex] is [tex]\( 10^3 = 1000 \)[/tex]:
[tex]\[
f(10^9) = -2 \cdot 1000 = -2000
\][/tex]
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\(-\infty \)[/tex].
2. As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \rightarrow -\infty \)[/tex]):
- Consider a very large negative value of [tex]\( x \)[/tex], such as [tex]\( -10^9 \)[/tex].
- Substituting [tex]\( x = -10^9 \)[/tex] into the function:
[tex]\[
f(-10^9) = -2 \left( -10^9 \right)^{1/3}
\][/tex]
- The cube root of [tex]\( -10^9 \)[/tex] is [tex]\( -1000 \)[/tex]:
[tex]\[
f(-10^9) = -2 \cdot (-1000) = 2000
\][/tex]
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( \infty \)[/tex].
Thus, the end behavior of the function [tex]\( f(x) = -2 \sqrt[3]{x} \)[/tex] is summarized as follows:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
The correct statement about the end behavior of the function 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].
