Question

the function f(x) is defined below. what is the end behavior of f(x)?

f(x)=-20 - 12x^2+4x^3 - 36x

answer attempt 1 out of 2

the degree of the polynomial is odd, and the leading coefficient is

choose which pair of limits below represents the end behavior:

lim_{x\to -\infty}f(x)=\infty, lim_{x\to\infty}f(x)=\infty

lim_{x\to -\infty}f(x)=-\infty, lim_{x\to\infty}f(x)=\infty

lim_{x\to -\infty}f(x)=\infty, lim_{x\to\infty}f(x)=-\infty

lim_{x\to -\infty}f(x)=-\infty, lim_{x\to\infty}f(x)=-\infty

Explanation:

Step1: Identify degree and leading coefficient

The polynomial $f(x)=-20 - 12x^{2}+4x^{3}-36x$ has degree 3 (highest - power of $x$) which is odd, and the leading coefficient is 4 (coefficient of $x^{3}$) which is positive.

Step2: Recall end - behavior rules

For a polynomial $y = a_nx^n+\cdots+a_0$ with $n$ odd and $a_n>0$, as $x\to-\infty$, $y\to-\infty$ and as $x\to\infty$, $y\to\infty$. That is, $\lim_{x\to-\infty}f(x)=-\infty$ and $\lim_{x\to\infty}f(x)=\infty$.

Answer:

The leading coefficient is positive. The pair of limits representing the end - behavior is $\lim_{x\to-\infty}f(x)=-\infty,\lim_{x\to\infty}f(x)=\infty$