Problem 8 - College Algebra Essentials

Question

Find the domain of each function. $$g(x)=\frac{2}{x^{2}+x-12}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x)=\frac{2}{x^{2}+x-12}\) is all real numbers except 3 and -4.

Step 1: Set the denominator equal to zero

First, set the equation \(x^{2}+x-12=0\). This will help us find which x-values we need to exclude from our domain, since these would make the function undefined.

Step 2: Solve for x

We can solve this quadratic equation by either factoring, completing the square, or using the quadratic formula. Since this is a simple quadratic and is able to be factored, we will use factoring: \(x^{2}+x-12=(x-3)(x+4)=0\). Setting each factor equal to zero gives us \(x=3\) and \(x=-4\).

Step 3: Write the domain

The domain of a function g(x) is all real numbers, except those that make the function undefined. In this case, \(x=3\) and \(x=-4\) cause the denominator to become 0 which would make the function undefined. Therefore, all other real numbers are part of the domain. The domain is \(x\in \mathbb{R}\) such that \(x\neq 3\) and \(x\neq -4\).