Given the functions \(f(x) = \frac{1}{x+2}\) and \(g(x) = x^2 - 4\), find the domain of the product of the functions \(h(x) = f(x)g(x)\).
Answer
Step 3: The domain of the product of the functions is the intersection of the domains of each of the functions. So, it's the intersection of \(x \in (-\infty, -2) \cup (-2, +\infty)\) and \(x \in (-\infty, +\infty)\).
Step 1 ：Step 1: First determine the domain of each function separately. For \(f(x)\), the denominator cannot be 0, so \(x+2 \neq 0\), which leads to \(x \neq -2\). So the domain of \(f(x)\) is \(x \in (-\infty, -2) \cup (-2, +\infty)\).
Step 2 ：Step 2: For \(g(x)\), since it is a quadratic function, its domain is all real numbers, i.e., \(x \in (-\infty, +\infty)\).
Step 3 ：Step 3: The domain of the product of the functions is the intersection of the domains of each of the functions. So, it's the intersection of \(x \in (-\infty, -2) \cup (-2, +\infty)\) and \(x \in (-\infty, +\infty)\).
