Problem 18 - College Algebra Essentials

Question

Find the domain of each function. $$f(x)=\sqrt{x+2}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\sqrt{x+2}\) is \([-2, \infty)\).

Step 1: Understand the function's domain restriction

For any function involving a square root, the argument of the square root (the expression under the square root symbol) must be greater than or equal to 0. This is because negative numbers do not have real square roots, only imaginary ones. So in this case, \(x + 2\) must be greater than or equal to 0.

Step 2: Solve the inequality

Take the inequality \(x + 2 \geq 0\) and solve it for x. To do this, subtract 2 from both sides of the inequality to isolate x. The solution becomes \(x \geq -2\).

Step 3: Express the domain

Now that we know the inequality that x must satisfy, the domain can be written in interval notation. The domain of the function includes all x-values greater than or equal to -2. Therefore, the domain in interval notation is \([-2, \infty)\).