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Chapter 2: Problem 12

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \(x = -2\).
03

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 \(x \in \mathbb{R}, x \neq 2, -2\)

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