Let \(f(x) = \sqrt{x+2}\) and \(g(x) = \frac{1}{x}\). Find the domain of the product of the functions \(f(x)g(x)\).
Answer
The intersection of \([-2, \infty)\) and \((-\infty, 0) \cup (0, \infty)\) is \((-2, 0) \cup (0, \infty)\).
Step 1 ：First, find the domain of each function separately. The function \(f(x)\) is defined for all \(x\) such that \(x+2 \geq 0\), so the domain of \(f(x)\) is \([-2, \infty)\). The function \(g(x)\) is defined for all \(x\) such that \(x \neq 0\), so the domain of \(g(x)\) is \((-\infty, 0) \cup (0, \infty)\).
Step 2 ：The product of the functions \(f(x)g(x)\) is defined where both functions are defined. This is the intersection of the domains of \(f(x)\) and \(g(x)\).
Step 3 ：The intersection of \([-2, \infty)\) and \((-\infty, 0) \cup (0, \infty)\) is \((-2, 0) \cup (0, \infty)\).
