complete the statements about the key features of the graph of (f(x)=x^{5}-9x^{3}). as (x) goes to negative…

Explanation:

Step1: Analyze the leading - term

The leading - term of the polynomial function (f(x)=x^{5}-9x^{3}) is (x^{5}). The general form of a polynomial is (a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0), and for large values of (|x|), the function behaves like its leading - term.

Step2: Consider (x\to-\infty)

When (x\to-\infty), for the function (y = x^{5}), if we let (x=-t) where (t\to+\infty), then (y=(-t)^{5}=-t^{5}). So as (x\to-\infty), (y = x^{5}\to-\infty). Since (f(x)) behaves like (x^{5}) for large (|x|), as (x\to-\infty), (f(x)\to-\infty).

Step3: Consider (x\to+\infty)

When (x\to+\infty), for the function (y = x^{5}), as (x) gets larger and larger, (y=x^{5}) also gets larger and larger. So as (x\to+\infty), (f(x)\to+\infty).

Answer:

As (x) goes to negative infinity, (f(x)) goes to negative infinity, and as (x) goes to positive infinity, (f(x)) goes to positive infinity.