To describe the end behavior of the function [tex]\( F(x) = -x^5 + x^2 - x \)[/tex], we need to analyze the term with the highest power, which determines the function's end behavior as [tex]\( x \)[/tex] approaches positive and negative infinity.
1. Identify the Leading Term:
The leading term of the function [tex]\( F(x) = -x^5 + x^2 - x \)[/tex] is [tex]\( -x^5 \)[/tex] because it has the highest power of [tex]\( x \)[/tex].
2. Analyze the Leading Term's Behavior:
- When [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to +\infty \)[/tex]):
The leading term [tex]\( -x^5 \)[/tex] will dominate the behavior of the function. Since the coefficient of [tex]\( x^5 \)[/tex] is negative, the term [tex]\( -x^5 \)[/tex] will approach negative infinity. Hence, [tex]\( F(x) \to -\infty \)[/tex] as [tex]\( x \to +\infty \)[/tex].
- When [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]):
Similarly, for large negative values of [tex]\( x \)[/tex], the negative sign and the odd power of [tex]\( x \)[/tex] means that [tex]\( -x^5 \)[/tex] will be a large positive number. Therefore, [tex]\( -x^5 \to +\infty \)[/tex]. Hence, [tex]\( F(x) \to +\infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
3. Determine the Graph's End Behavior:
Based on the above analysis, the graph of [tex]\( F(x) = -x^5 + x^2 - x \)[/tex]:
- Starts from positive infinity as [tex]\( x \)[/tex] approaches negative infinity (the left side of the graph).
- Ends at negative infinity as [tex]\( x \)[/tex] approaches positive infinity (the right side of the graph).
Thus, the correct end behavior of the graph is:
Option A: The graph of the function starts high and ends low.
