Question

consider the following function:

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

what is the domain of the function?

Explanation:

Identify the condition for the domain

Using the Domain of Rational Functions knowledge point, a rational function is defined for all real numbers except where its denominator is equal to zero.

$$ 3x^2 - 3 eq 0 $$

Solve for the excluded values

We find the values of \(x\) that make the denominator zero by factoring the quadratic expression.

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

$$ 3(x - 1)(x + 1) = 0 $$

$$ x = 1 \quad \text{or} \quad x = -1 $$

Express the domain in interval notation

Since \(x\) cannot be \(-1\) or \(1\), we exclude these two points from the set of all real numbers. This divides the real number line into three separate intervals:

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

Answer:

Consider the following function:

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

What is the domain of the function?
<blank>\((-\infty, -1) \cup (-1, 1) \cup (1, \infty)\)</blank>