The function you provided is:

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

### To analyze this function, let's break it down:

1. **Domain**:  
   The function is defined for all real values of \( x \) because the denominator \( x^2 + 4 \) is never zero (since \( x^2 + 4 \geq 4 \) for all real \( x \)).

2. **Simplification**:  
   The expression is already in its simplest form. There is no factor that cancels between the numerator and denominator.

3. **Asymptotes**:
   - **Horizontal Asymptote**: 
     As \( x \to \infty \), the highest degree term in the denominator \( x^2 \) dominates the function. So,
     \[
     \lim_{x \to \infty} \frac{8x}{x^2 + 4} = \lim_{x \to \infty} \frac{8}{x} = 0
     \]
     Thus, the horizontal asymptote is \( y = 0 \).

   - **Vertical Asymptote**:  
     Since the denominator \( x^2 + 4 \) is never zero, there are no vertical asymptotes.

4. **Intercepts**:
   - **x-intercept**: Set \( f(x) = 0 \) and solve for \( x \):
     \[
     \frac{8x}{x^2 + 4} = 0
     \]
     The numerator must be zero, so \( 8x = 0 \) implies \( x = 0 \). Thus, the x-intercept is \( (0, 0) \).
   
   - **y-intercept**: This is the same as the x-intercept since \( f(0) = 0 \).

5. **Behavior at Infinity**:
   As \( x \to \infty \) or \( x \to -\infty \), the function approaches the horizontal asymptote \( y = 0 \).

### Summary:
- Domain: All real numbers \( \mathbb{R} \)
- Horizontal Asymptote: \( y = 0 \)
- No Vertical Asymptote
- Intercept: \( (0, 0) \)

Would you like to go deeper into any part of this explanation? Or plot the graph?

Here are 5 related questions to extend this analysis:
1. What is the derivative of the function \( f(x) = \frac{8x}{x^2 + 4} \)?
2. How do we find the critical points of the function \( f(x) \)?
3. What is the second derivative of \( f(x) \) and what does it tell us about concavity?
4. Can we analyze the behavior of this function as \( x \to -\infty \)?
5. How would we integrate \( f(x) = \frac{8x}{x^2 + 4} \)?

**Tip**: Always check the domain of a rational function by ensuring the denominator is never zero.

Explore the detailed analysis of the rational function f(x) = 8x/(x^2 + 4), including domain, horizontal and vertical asymptotes, intercepts, and behavior at infinity. Ideal for high school students in Grades 10-12.

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