The given function is:

$y = \frac{e^{2x}}{e^x + 3}$

This is a rational function involving exponential terms. How would you like to proceed with it? For instance, we could:

1. Find the first derivative ($y'$) using differentiation.
2. Analyze the behavior of the function, such as finding asymptotes or critical points.
3. Determine the limits as $x \to \infty$ or $x \to -\infty$.
4. Find the second derivative to analyze concavity.
5. Solve for $x$ when $y = 0$ or another specific value.

Would you like details on any of these, or do you have a different request for this function?

Also, here are some questions that could help guide further exploration:

1. What is the derivative of $y = \frac{e^{2x}}{e^x + 3}$?
2. What are the critical points of this function?
3. What is the behavior of the function as $x \to \infty$?
4. Can we simplify the given function in any way?
5. What is the second derivative of the function?

Tip: When differentiating a quotient, always apply the quotient rule: $\frac{f}{g}$ becomes $\frac{f'g - fg'}{g^2}$.