In Exercises 1–30, find the domain of each function. h(x) = √(x −2)+ √(x +3)

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1. Identify the conditions under which the square root function is defined. The expression under the square root must be greater than or equal to zero.
2. For the first square root, \( \sqrt{x - 2} \), set up the inequality: \( x - 2 \geq 0 \).
3. Solve the inequality \( x - 2 \geq 0 \) to find the values of \( x \) that satisfy it.
4. For the second square root, \( \sqrt{x + 3} \), set up the inequality: \( x + 3 \geq 0 \).
5. Solve the inequality \( x + 3 \geq 0 \) to find the values of \( x \) that satisfy it. The domain of \( h(x) \) is the intersection of the solutions to these inequalities.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For functions involving square roots, the expression inside the root must be non-negative, as the square root of a negative number is not defined in the set of real numbers.

Square Root Function

A square root function is defined as f(x) = √x, where x must be greater than or equal to zero. This means that for any expression under the square root, it must satisfy the condition that the expression is non-negative to ensure the function yields real number outputs.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values, indicating that one is greater than, less than, or equal to the other. In finding the domain of the function h(x), we will set up inequalities based on the conditions required for the square roots to be defined, allowing us to determine the valid range of x-values.