Question

the function f(x) is defined below. what is the end behavior of f(x)?
f(x)=-35x + 42x^3-574x^2 + 525-7x^5 + 49x^4
answer attempt 1 out of 2
the degree of the polynomial is, and the leading coefficient is
choose which pair of limits below represents the end behavior:
lim_{x->-∞} f(x)=∞, lim_{x->∞} f(x)=∞
lim_{x->-∞} f(x)=-∞, lim_{x->∞} f(x)=∞
lim_{x->-∞} f(x)=∞, lim_{x->∞} f(x)=-∞
lim_{x->-∞} f(x)=-∞, lim_{x->∞} f(x)=-∞

Explanation:

Step1: Identify degree of polynomial

The degree of a polynomial is the highest - power of the variable. For \(f(x)=-7x^{5}+49x^{4}+42x^{3}-574x^{2}-35x + 525\), the degree is \(5\) (an odd number).

Step2: Identify leading coefficient

The leading coefficient is the coefficient of the term with the highest - power of the variable. Here, the leading coefficient of \(f(x)\) is \(-7\) (a negative number).

Step3: Determine end - behavior

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 degree of the polynomial is odd, and the leading coefficient is \(-7\). The pair of limits that represents the end - behavior is \(\lim_{x\to-\infty}f(x)=\infty\), \(\lim_{x\to\infty}f(x)=-\infty\)