Let's analyze the function [tex]\( f(x) = 3 \sqrt[3]{x} \)[/tex] step-by-step to determine its end behavior.
1. Understanding the Cube Root Function:
- The cube root function, [tex]\( \sqrt[3]{x} \)[/tex], is defined for all real numbers.
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], the cube root of a negative number remains negative and continues to decrease without bound.
- As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], the cube root of a positive number remains positive and increases without bound.
2. Multiplying by 3:
- The function [tex]\( f(x) = 3 \sqrt[3]{x} \)[/tex] is three times the cube root of [tex]\( x \)[/tex].
- Multiplying by 3 stretches the values of the cube root function but does not change its end behavior.
3. End Behavior Analysis:
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( \sqrt[3]{x} \)[/tex] approaches [tex]\( -\infty \)[/tex]. Therefore, [tex]\( 3 \sqrt[3]{x} \)[/tex] also approaches [tex]\( -\infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( \sqrt[3]{x} \)[/tex] approaches [tex]\( \infty \)[/tex]. Therefore, [tex]\( 3 \sqrt[3]{x} \)[/tex] also approaches [tex]\( \infty \)[/tex].
Combining these observations, we conclude that:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
Thus, the correct answer is:
- As [tex]\( x \to -\infty, f(x) \to -\infty \)[/tex] and as [tex]\( x \to \infty, f(x) \to \infty \)[/tex].
The accurate statement representing the end behavior of [tex]\( f(x) = 3 \sqrt[3]{x} \)[/tex] is:
- As [tex]\( x \rightarrow -\infty, f(x) \rightarrow -\infty \)[/tex], and as [tex]\( x \rightarrow \infty, f(x) \rightarrow \infty \)[/tex].
