select the correct answer.\nwhich pair of statements describes the end - behavior of the graph of the…

select the correct answer.\nwhich pair of statements describes the end - behavior of the graph of the function (f(x)=x^{3}+2x^{2}-5x - 6)?\na. as (x) approaches negative infinity, (f(x)) approaches infinity. as (x) approaches infinity, (f(x)) approaches infinity.\nb. as (x) approaches negative infinity, (f(x)) approaches infinity. as (x) approaches infinity, (f(x)) approaches negative infinity.\nc. as (x) approaches negative infinity, (f(x)) approaches negative infinity. as (x) approaches infinity, (f(x)) approaches negative infinity.\nd. as (x) approaches negative infinity, (f(x)) approaches negative infinity. as (x) approaches infinity, (f(x)) approaches infinity.

select the correct answer.\nwhich pair of statements describes the end - behavior of the graph of the function (f(x)=x^{3}+2x^{2}-5x - 6)?\na. as (x) approaches negative infinity, (f(x)) approaches infinity. as (x) approaches infinity, (f(x)) approaches infinity.\nb. as (x) approaches negative infinity, (f(x)) approaches infinity. as (x) approaches infinity, (f(x)) approaches negative infinity.\nc. as (x) approaches negative infinity, (f(x)) approaches negative infinity. as (x) approaches infinity, (f(x)) approaches negative infinity.\nd. as (x) approaches negative infinity, (f(x)) approaches negative infinity. as (x) approaches infinity, (f(x)) approaches infinity.

Answer

Explanation:

Step1: Identify the leading - term

The function is $f(x)=x^{3}+2x^{2}-5x - 6$, and the leading - term is $x^{3}$ (the term with the highest power of $x$).

Step2: Analyze the end - behavior for $x\to-\infty$

When $x\to-\infty$, for the leading - term $y = x^{3}$, if we substitute a very large negative number for $x$, say $x=-N$ where $N$ is a large positive number, then $y=(-N)^{3}=-N^{3}\to-\infty$. So as $x\to-\infty$, $f(x)\to-\infty$ since the leading - term dominates the behavior of the polynomial for large values of $|x|$.

Step3: Analyze the end - behavior for $x\to\infty$

When $x\to\infty$, for the leading - term $y = x^{3}$, if we substitute a very large positive number for $x$, say $x = N$ where $N$ is a large positive number, then $y=N^{3}\to\infty$. So as $x\to\infty$, $f(x)\to\infty$ since the leading - term dominates the behavior of the polynomial for large values of $|x|$.

Answer:

D. As $x$ approaches negative infinity, $f(x)$ approaches negative infinity. As $x$ approaches infinity, $f(x)$ approaches infinity.