To determine the end behavior of the function [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex], we analyze its leading term, which is [tex]\( x^3 \)[/tex] in this case. The behavior of the function as [tex]\( x \)[/tex] approaches positive or negative infinity is dictated by this leading term.
For the cubic polynomial [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex]:
1. As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]):
- The leading term [tex]\( x^3 \)[/tex] dominates the behavior.
- When [tex]\( x \)[/tex] is a large negative number, [tex]\( x^3 \)[/tex] becomes a large negative number.
- Therefore, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
2. As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to \infty \)[/tex]):
- The leading term [tex]\( x^3 \)[/tex] again dominates the behavior.
- When [tex]\( x \)[/tex] is a large positive number, [tex]\( x^3 \)[/tex] becomes a large positive number.
- Therefore, as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
Based on this analysis, the correct end behaviors described are:
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
- As [tex]\( x \)[/tex] approaches infinity, [tex]\( f(x) \)[/tex] approaches infinity.
The correct answer is:
D. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity. As [tex]\( x \)[/tex] approaches infinity, \( f(x) approaches infinity.
