To determine the end behavior of the polynomial function [tex]\( g(x) = 10x^9 - 50x^6 - 500x^2 \)[/tex], we need to focus on the leading term, which is the term with the highest power of [tex]\( x \)[/tex].
For the polynomial [tex]\( g(x) \)[/tex], the leading term is [tex]\( 10x^9 \)[/tex]. The coefficient of this term is [tex]\( 10 \)[/tex], a positive number, and the exponent [tex]\( 9 \)[/tex] is an odd number.
Now, let's analyze the end behavior based on the leading term:
1. As [tex]\( x \rightarrow \infty \)[/tex]:
- For very large positive values of [tex]\( x \)[/tex], the term [tex]\( 10x^9 \)[/tex] will dominate because it grows much faster than any other terms in the polynomial.
- Given that [tex]\( x \)[/tex] is positive and the coefficient [tex]\( 10 \)[/tex] is also positive, [tex]\( 10x^9 \)[/tex] will grow positively without bound.
- Therefore, as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( g(x) \rightarrow \infty \)[/tex].
2. As [tex]\( x \rightarrow -\infty \)[/tex]:
- For very large negative values of [tex]\( x \)[/tex], the term [tex]\( 10x^9 \)[/tex] still dominates.
- Since [tex]\( x \)[/tex] is negative and the exponent 9 is odd, [tex]\( x^9 \)[/tex] will be negative, and hence [tex]\( 10x^9 \)[/tex] will be a large negative number.
- As a result, [tex]\( 10x^9 \)[/tex] will grow negatively without bound.
- Therefore, as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( g(x) \rightarrow -\infty \)[/tex].
Given this analysis, the correct end behavior is:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( g(x) \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( g(x) \rightarrow -\infty \)[/tex].
Thus, the correct answer is:
(D) As [tex]\( x \rightarrow \infty, g(x) \rightarrow \infty \)[/tex], and as [tex]\( x \rightarrow -\infty, g(x) \rightarrow -\infty \)[/tex].
