Problem

Find the domain of the difference of the functions \(f(x) = \sqrt{x - 1}\) and \(g(x) = \frac{1}{x + 2}\).

Solution

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 \(x \geq 1\).

From Solvely APP
Source: https://solvelyapp.com/problems/2QqUSKDuEw/

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