Questions: Find the domain of the function. g(x) = 2x / (x^2 - 9)

Solution Steps

To find the domain of the function \( g(x) = \frac{2x}{x^2 - 9} \), we need to determine the values of \( x \) for which the function is defined. The function is undefined when the denominator is zero. Therefore, we need to find the values of \( x \) that make \( x^2 - 9 = 0 \) and exclude them from the domain.

We are given the function \( g(x) = \frac{2x}{x^2 - 9} \). To find the domain, we need to determine where this function is defined.

The function is undefined when the denominator is equal to zero. Thus, we set the denominator equal to zero: \[ x^2 - 9 = 0 \]

Solving the equation \( x^2 - 9 = 0 \) gives us: \[ x^2 = 9 \implies x = \pm 3 \] This means the function is undefined at \( x = -3 \) and \( x = 3 \).

The domain of \( g(x) \) excludes the points where the function is undefined. Therefore, the domain can be expressed in interval notation as: \[ (-\infty, -3) \cup (-3, 3) \cup (3, \infty) \]

Final Answer