Question

WILL MARK BRAINLIEST IF CORRECT!

Answer 1

Answer:

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

Step-by-step explanation:

$f(x)=-x^4-x^3+3x^2+1$

The question wants you to determine the end behavior of this polynomial function.

In order to determine the end behavior of polynomial functions, you want to look at the polynomial in its general form. This means that the variables have powers that are in descending order.

Look at the leading term of the polynomial in general form. Here, the function $f(x)=-x^4-x^3+3x^2+1$ is in general form, so we can look at the first term (term with the highest power).

The leading term for this polynomial function is $-x^4$. The leading term has the highest power; therefore, its behavior will dominate the graph shape.

When graphed, this function will behave similarly to an upside down parabola.

Based on the leading term, we can see that the leading coefficient is -1 and the degree is 4.

Look to see if the degree is even/odd, and if the leading coefficient is negative/positive.

The degree of 4 is even, and the leading coefficient of -1 is negative.

When the degree is even and the LC is negative, the graph of the polynomial function falls to the left and falls to the right.

Therefore, if the end behavior falls to the left and falls to the right, that means that as x approaches negative infinity and positive infinity, the y-values will approach negative infinity and negative infinity on either side of the graph.

The answer is the rightmost option:

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