Question

multiple - choice question
for the function f, lim(x→ - ∞) f(x) = ∞ and lim(x→+∞) f(x)= - ∞, which of the following must be true about f?
the degree of f is even, and the leading coefficient is negative.
the degree of f is odd, and the leading coefficient is negative.
the degree of f is odd, and the leading coefficient is positive.
the degree of f is even, and the leading coefficient is positive.

Explanation:

Step1: Recall end - behavior rules

For a polynomial function $f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_0$, the end - behavior is determined by the degree $n$ and the leading coefficient $a_n$.

Step2: Analyze cases for different degrees and leading coefficients

If $n$ is even: $\lim_{x
ightarrow-\infty}f(x)=\lim_{x
ightarrow\infty}f(x)$ (both $\infty$ or both $-\infty$) depending on the sign of $a_n$. If $n$ is odd: $\lim_{x
ightarrow-\infty}f(x)=-\lim_{x
ightarrow\infty}f(x)$. Given $\lim_{x
ightarrow-\infty}f(x)=\infty$ and $\lim_{x
ightarrow\infty}f(x)=-\infty$, the degree $n$ must be odd. When $n$ is odd and $a_n<0$, $\lim_{x
ightarrow-\infty}f(x)=\infty$ and $\lim_{x
ightarrow\infty}f(x)=-\infty$.

Answer:

The degree of $f$ is odd, and the leading coefficient is negative.