It seems you've shared an image related to a rational function, and you want to determine its domain. Let's break it down.

The function is:

$f(x) = \frac{2x}{3x^2 - 3}$

To find the domain, we need to determine for which values of $x$ the function is defined. A rational function is undefined when the denominator equals zero because division by zero is not possible.

So, let's set the denominator equal to zero and solve for $x$:

$3x^2 - 3 = 0$

Solve for $x$:

$3(x^2 - 1) = 0$ $x^2 - 1 = 0$ $x^2 = 1$ $x = \pm 1$

Thus, the function is undefined when $x = 1$ and $x = -1$.

Therefore, the domain of the function is all real numbers except $x = 1$ and $x = -1$, which can be written in interval notation as:

$(-\infty, -1) \cup (-1, 1) \cup (1, \infty)$

Would you like further clarification or additional steps related to this function? Here are a few questions that could help deepen your understanding:

1. How do vertical asymptotes relate to the domain of a rational function?
2. What happens if the numerator of the rational function equals zero?
3. How would the graph of this function behave near $x = 1$ and $x = -1$?
4. How do you find horizontal asymptotes in rational functions?
5. Can you apply this approach to a more complex rational function?

Tip: Always check for values that make the denominator zero first when analyzing the domain of a rational function.