Problem 8 - Precalculus 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 \(-4\) and \(3\).

Step 1: Setting the Denominator Equal to Zero

To find the values of \(x\) for which the denominator equals zero, we set \(x^{2} + x - 12 = 0\)

Step 2: Solving the Quadratic Equation

Solving the quadratic equation, we can factorize it as \((x-3)(x+4)=0\). This gives two solutions: \(x = -4\) and \(x = 3\). These are the values of \(x\) at which the denominator is zero.

Step 3: Determining the Domain

Since the denominator cannot be zero, the domain of \(g(x)\) is all real numbers except the values that make the denominator equal to zero, which we found to be \(x = -4\) and \(x = 3\).