Find the domain of the difference of the functions \(f(x) = \sqrt{x - 1}\) and \(g(x) = \frac{1}{x + 2}\).
Answer
Therefore, the domain of \(f(x) - g(x)\) is the set of all \(x\) such that \(x \geq 1\) and \(x \neq -2\). Since \(1 > -2\), the second condition is automatically satisfied when \(x \geq 1\).
Step 1 ：To find the domain of the difference of the functions \(f(x) - g(x)\), we need to find the set of all real numbers that are in both the domain of \(f(x)\) and the domain of \(g(x)\).
Step 2 ：The function \(f(x) = \sqrt{x - 1}\) is defined for all \(x\) such that \(x - 1 \geq 0\). Solving this inequality gives \(x \geq 1\).
Step 3 ：The function \(g(x) = \frac{1}{x + 2}\) is defined for all \(x\) such that \(x + 2 \neq 0\). Solving this equation gives \(x \neq -2\).
Step 4 ：Therefore, the domain of \(f(x) - g(x)\) is the set of all \(x\) such that \(x \geq 1\) and \(x \neq -2\). Since \(1 > -2\), the second condition is automatically satisfied when \(x \geq 1\).
