To find the domain of the function $\frac{2x^2}{x-3}$, we need to determine for which values of $x$ the function is undefined.

The function is undefined wherever the denominator is zero, as division by zero is not allowed. The denominator of the function is $x - 3$. To find the critical value, we solve:

$x - 3 = 0$ $x = 3$

Therefore, the function is undefined at $x = 3$, meaning $x = 3$ is not in the domain.

Conclusion:

The domain of the function $\frac{2x^2}{x-3}$ is all real numbers except $x = 3$.

In interval notation, the domain is: $(-\infty, 3) \cup (3, \infty)$

Would you like further details or examples? Here are some related questions:

1. How do we find the domain of rational functions with multiple factors in the denominator?
2. What is the behavior of $\frac{2x^2}{x-3}$ as $x$ approaches 3 from the left and right?
3. How can we simplify the expression $\frac{2x^2}{x-3}$ if needed?
4. What happens if the denominator has higher-degree polynomials?
5. How do we check for vertical asymptotes in rational functions?

Tip: Always check for values of $x$ that make the denominator zero to avoid undefined points in rational functions.