Problem 25 - College Algebra Essentials

Question

Find the domain of each function. $$h(x)=\sqrt{x-2}+\sqrt{x+3}$$

Step-by-Step Solution

Answer

The domain of the function \(h(x)=\sqrt{x-2}+\sqrt{x+3}\) is \(x \geq 2\).

Step 1: Setting each Root Greater or Equal to Zero

To find the domain, let's set the expression under each square root greater or equal to zero. So we start by setting \(x-2 \geq 0\) and \(x+3 \geq 0\).

Step 2: Solving Inequalities

Solving these inequalities results in two solutions for x: \(x \geq 2\) from \(x-2 \geq 0\) and \(x \geq -3\) from \(x+3 \geq 0\).

Step 3: Intersection of Solution

Since both conditions have to be satisfied (i.e., both square roots need to be defined), the final domain is the intersection of both solution sets, which means taking the greatest lower limit. Our domain is then \(x \geq 2\).