To find the behavior of the function \( y = \frac{x^{3} + x}{3x^{2} - 4} \), let's analyze it step by step.

### 1. Domain:
The function \( y = \frac{x^{3} + x}{3x^{2} - 4} \) is defined for all \( x \) where the denominator \( 3x^{2} - 4 \neq 0 \). 

So, let's solve for \( x \):
\[
3x^{2} - 4 = 0 \implies x^{2} = \frac{4}{3} \implies x = \pm \frac{2}{\sqrt{3}} = \pm \frac{2\sqrt{3}}{3}
\]

Thus, the function is undefined at \( x = \pm \frac{2\sqrt{3}}{3} \).

### 2. Simplification:
There is no further simplification possible for the expression.

### 3. Asymptotes:
- **Vertical Asymptotes**: Occur where the denominator is zero (as discussed above). Therefore, vertical asymptotes are at \( x = \pm \frac{2\sqrt{3}}{3} \).
- **Horizontal Asymptote**: To find the horizontal asymptote, we consider the limit as \( x \) approaches infinity or negative infinity.
  
  For large values of \( x \):
  \[
  y \approx \frac{x^3}{3x^2} = \frac{x}{3}
  \]
  Therefore, as \( x \) approaches infinity or negative infinity, \( y \) also increases or decreases without bound. Thus, there is no horizontal asymptote.

### 4. Derivative (Critical Points and Inflection Points):
To find the critical points, we take the derivative using the quotient rule.

Given \( y = \frac{x^{3} + x}{3x^{2} - 4} \), let:
- \( u = x^{3} + x \)
- \( v = 3x^{2} - 4 \)

The derivative \( y' \) is given by:
\[
y' = \frac{v \cdot u' - u \cdot v'}{v^2}
\]
Where:
- \( u' = 3x^{2} + 1 \)
- \( v' = 6x \)

Substitute \( u \), \( u' \), \( v \), and \( v' \) into the derivative formula:
\[
y' = \frac{(3x^2 - 4)(3x^2 + 1) - (x^3 + x)(6x)}{(3x^2 - 4)^2}
\]

Simplify to find the critical points and determine concavity.

### 5. Graphical Behavior:
Analyzing the derivative further helps us understand the behavior of the graph, including increasing/decreasing intervals and concavity.

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Would you like to see the detailed calculation of the derivative or have any other specific questions?

### Related Questions:
1. What are the critical points and how do they affect the function's graph?
2. How do you determine intervals of increase and decrease using the derivative?
3. Can we find the points of inflection for this function?
4. How does the function behave near its vertical asymptotes?
5. What is the significance of concavity in graphing functions?

**Tip:** When dealing with rational functions, always check where the denominator equals zero to identify points where the function is undefined.

Explore the detailed analysis of the rational function y = (x^3 + x) / (3x^2 - 4). This problem covers key calculus concepts such as derivatives, asymptotes, and the behavior of rational functions. Suitable for students in Grades 11-12.

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