Problem 7 - Precalculus Essentials

Question

Find the domain of each function. $$g(x)=\frac{3}{x^{2}-2 x-15}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \frac{3}{{x^{2}-2x-15}}\) is all real numbers, except \(x \neq 5\) and \(x \neq -3\).

Step 1: Identify the Function Type

The provided function \(g(x) = \frac{3}{{x^{2}-2x-15}}\) is a rational function where the numerator and denominator are polynomials. The denominator cannot be zero because division by zero is undefined in mathematics.

Step 2: Find Values that Makes the Denominator Zero

To determine the restrictions on the domain, we need to identify the values that make the denominator zero. Therefore, we solve the equation \(0 = x^{2} - 2x - 15\). This equation can be factored to become \(0 = (x-5)(x+3)\). Thus, \(x = 5\) and \(x = -3\) make the denominator zero.

Step 3: Define the Domain

The domain of a function is the set of all values of \(x\) for which the function is defined. For this function, it is defined for all real numbers except for the points where the denominator is zero. So, the domain of \(g(x)\) is \(x \in \mathbb{R}\), except \(x \neq 5\) and \(x \neq -3\).