The given function is:

$f(x) = \frac{8x}{x^2 + 4}$

Simplifying the Function

At first glance, the function is already in its simplest form. The numerator is $8x$, and the denominator is $x^2 + 4$. This is a rational function, and you can't simplify it any further.

Domain of the Function

To find the domain of the function, we look for values of $x$ that would make the denominator zero. In this case, $x^2 + 4 = 0$ leads to:

$x^2 = -4$

Since there are no real numbers whose square is negative, the denominator is never zero for any real $x$. Therefore, the domain of $f(x)$ is all real numbers:

$\text{Domain: } (-\infty, \infty)$

Asymptotes

• Vertical Asymptotes: Since the denominator $x^2 + 4$ is never zero, there are no vertical asymptotes.
• Horizontal Asymptote: As $x \to \infty$ or $x \to -\infty$, the function behaves like $\frac{8x}{x^2} = \frac{8}{x}$, which approaches 0. Therefore, the horizontal asymptote is $y = 0$.

Behavior of the Function

• At $x = 0$:

$f(0) = \frac{8(0)}{0^2 + 4} = 0$

• As $x \to \infty$ or $x \to -\infty$: As mentioned, the function tends towards 0, suggesting a horizontal asymptote at $y = 0$.

Would you like further details or any specific clarification?

Related Questions:

1. What are the critical points and extrema of the function $f(x)$?
2. How do you find the derivative $f'(x)$ of the function?
3. Can you find the concavity of $f(x)$ by calculating the second derivative $f''(x)$?
4. What is the range of the function $f(x)$?
5. How do you sketch the graph of $f(x)$ with all key points?

Tip: When analyzing a rational function, always check for vertical and horizontal asymptotes to understand its long-term behavior.