Question

the function $f(x)$ is defined below. what is the end behavior of $f(x)$?
$f(x) = x^4 - 72 + 5x^2 - 18x$
answer
as $x \to \infty, f(x) \to \infty$ and
as $x \to -\infty, f(x) \to \infty$
as $x \to \infty, f(x) \to -\infty$ and
as $x \to -\infty, f(x) \to \infty$
as $x \to \infty, f(x) \to -\infty$ and
as $x \to -\infty, f(x) \to -\infty$
as $x \to \infty, f(x) \to \infty$ and
as $x \to -\infty, f(x) \to -\infty$

Explanation:

Step1: Identify leading term

The leading term of $f(x) = x^4 + 5x^2 - 18x - 72$ is $x^4$.

Step2: Analyze degree and coefficient

The degree (4) is even, and the leading coefficient (1) is positive.

Step3: Determine end behavior

For even degree with positive leading coefficient:
As $x \to \infty$, $x^4 \to \infty$, so $f(x) \to \infty$.
As $x \to -\infty$, $(-x)^4 = x^4 \to \infty$, so $f(x) \to \infty$.

Answer:

as $x \to \infty, f(x) \to \infty$ and as $x \to -\infty, f(x) \to \infty$