Problem 12 - Precalculus 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 is all real numbers except \(x = 2\) and \(x = -2\). In interval notation, it can be represented as \((-\infty, -2) \cup (-2, 2) \cup (2, \infty)\).

Step 1: Set the first denominator equal to 0

Firstly, we need to find the x values that will make the first denominator zero. This means solving the equation \(x^{2} + 4 = 0\). Subtract 4 from both sides, we get \(x^{2} = -4\). However, for real numbers, there is no x that would make \(x^{2}\) negative. Thus, the first fraction has no undefined points.

Step 2: Set the second denominator equal to 0

Next, let us set the second denominator equal to zero. Solve the equation \(x^{2} - 4 = 0\). We add 4 to both sides and square root both sides to get \(x = 2\) and \(x = -2\). This implies that the function is undefined at \(x = 2\) and \(x = -2\).

Step 3: Determine the domain of the function

Now that we've identified the x values which make the function undefined, remove these points from the set of all real numbers to get the domain. Therefore, the domain of the function is all real numbers except \(x = 2\) and \(x = -2\). In mathematical notation, the domain is \(-\infty < x < -2\) or \(-2 < x < 2\) or \(2 < x < \infty\).