Questions: Find the domain of the following rational function: g(x) = (x - 2)(x + 3)/-2x^2

Transcript text: Find the domain of the following rational function: g(x) = \frac{(x - 2)(x + 3)}{-2x^2}

Solution

Solution Steps

To find the domain of the rational function \( g(x) = \frac{(x - 2)(x + 3)}{-2x^2} \), we need to determine the values of \( x \) for which the function is defined. The function is undefined where the denominator is zero. Therefore, we need to find the values of \( x \) that make the denominator zero and exclude them from the domain.

Step 1: Identify the Denominator

The given rational function is

\[ g(x) = \frac{(x - 2)(x + 3)}{-2x^2} \]

To find the domain, we first identify the denominator of the function, which is \(-2x^2\).

Step 2: Determine Undefined Points

The function is undefined where the denominator is zero. Therefore, we solve the equation:

\[ -2x^2 = 0 \]

Solving for \( x \), we find:

\[ x = 0 \]

Step 3: Define the Domain

The domain of the function consists of all real numbers except where the function is undefined. Since the function is undefined at \( x = 0 \), the domain is:

\[ (-\infty, 0) \cup (0, \infty) \]