Problem 12 - College Algebra Essentials

Question

Find the domain of each function. $$g(x)=\frac{1}{x^{2}+4}-\frac{1}{x^{2}-4}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x)\) is all real numbers except \(x = 2\) and \(x = -2\). So, \(x \in \mathbb{R}, x \neq 2, -2\).

Step 1: Analyze the denomenator of each fraction

The two fractions in the function have these denominators: \(x^{2}+4\) and \(x^{2}-4\). In general, the domain of a function will include all real numbers except where the denominator is zero.

Step 2: Identify where the denominator is zero

Set each denominator to zero and solve for \(x\). Doing this, we see that \(x^{2}+4 = 0\) will yield no real solution, meaning it will not limit the function's domain. However, \(x^{2}-4 = 0\) gives \(x = 2\) and \(x = -2\).

Step 3: Combine the results

Taking the results from step 2, we can say that the function is defined for every real number except \(x = 2\) and \(x = -2\). Hence, the domain of \(g(x)\) is \(x \in \mathbb{R}, x \neq 2, -2\)