Question

what is the end behavior of the function $f(x) = 8x^5 - 3x^3 + 8x + 5$?
a. as $x$ approaches infinity, $f(x)$ approaches negative infinity, and as $x$ approaches negative infinity, $f(x)$ approaches...
b. as $x$ approaches infinity, $f(x)$ approaches negative infinity, and as $x$ approaches negative infinity, $f(x)$ approaches negative infinity.
c. as $x$ approaches negative infinity, $f(x)$ approaches infinity, and as $x$ approaches infinity, $f(x)$ approaches negative infinity.
d. as $x$ approaches infinity, $f(x)$ approaches infinity, and as $x$ approaches negative infinity, $f(x)$ approaches negative infinity.

Explanation:

Step1: Identify leading term

The leading term of $f(x)=8x^5 - 3x^3 + 8x + 5$ is $8x^5$.

Step2: Analyze degree and leading coefficient

Degree is 5 (odd), leading coefficient $8>0$.

Step3: Determine end behavior

For odd degree, positive leading coefficient:
As $x\to\infty$, $8x^5\to\infty$, so $f(x)\to\infty$.
As $x\to-\infty$, $8x^5\to-\infty$, so $f(x)\to-\infty$.

Answer:

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