To understand the behavior of the function [tex]\( f(x) = x^5 - 9x^3 \)[/tex] as [tex]\( x \)[/tex] approaches negative infinity and positive infinity, we need to examine the dominant term in the function for very large positive and negative values of [tex]\( x \)[/tex].
1. As [tex]\( x \)[/tex] goes to negative infinity:
- When [tex]\( x \)[/tex] is a very large negative number, the term [tex]\( x^5 \)[/tex] (which is the highest degree term) will dominate the behavior of the function because it grows much faster than [tex]\( -9x^3 \)[/tex] as [tex]\( x \)[/tex] becomes more negative.
- Since [tex]\( x^5 \)[/tex] has an odd exponent, it will be negative when [tex]\( x \)[/tex] is negative. Therefore, as [tex]\( x \)[/tex] goes to negative infinity, the term [tex]\( x^5 \)[/tex] will drag [tex]\( f(x) \)[/tex] to negative infinity.
Hence, as [tex]\( x \)[/tex] goes to negative infinity, [tex]\( f(x) \)[/tex] goes to negative infinity.
2. As [tex]\( x \)[/tex] goes to positive infinity:
- When [tex]\( x \)[/tex] is a very large positive number, again, the term [tex]\( x^5 \)[/tex] will dominate because it is of the highest degree and grows faster than [tex]\( -9x^3 \)[/tex].
- For positive [tex]\( x \)[/tex], [tex]\( x^5 \)[/tex] is positive, and thus [tex]\( f(x) \)[/tex] will be driven towards positive infinity due to the [tex]\( x^5 \)[/tex] term.
Therefore, as [tex]\( x \)[/tex] goes to positive infinity, [tex]\( f(x) \)[/tex] goes to positive infinity.
Based on the above reasoning, the key features of the graph of [tex]\( f(x) = x^5 - 9x^3 \)[/tex] can be summarized as:
- As [tex]\( x \)[/tex] goes to negative infinity, [tex]\( f(x) \)[/tex] goes to negative infinity.
- As [tex]\( x \)[/tex] goes to positive infinity, [tex]\( f(x) \)[/tex] goes to positive infinity.
