Question

consider the following function:
$f(x) = \frac{2x}{3x^2 - 3}$
what is the domain of the function?

Explanation:

Step1: Identify denominator restriction

A rational function is undefined when its denominator equals 0. Set the denominator equal to 0:
$$3x^2 - 3 = 0$$

Step2: Solve for x

Factor and solve the equation:
$$3(x^2 - 1) = 0$$
$$x^2 - 1 = 0$$
$$x^2 = 1$$
$$x = \pm 1$$

Step3: Define valid x-values

The function is defined for all real numbers except $x=1$ and $x=-1$.

Answer:

All real numbers except $x=1$ and $x=-1$, or in interval notation: $(-\infty, -1) \cup (-1, 1) \cup (1, \infty)$