Find the domain of each function. f(x) = √(24 - 2x)

Verified step by step guidance

1. The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the square root function, the expression inside the square root must be greater than or equal to zero because the square root of a negative number is undefined in the real number system.
2. Start by setting the expression inside the square root greater than or equal to zero: 24 - 2x ≥ 0.
3. Solve the inequality for x. First, isolate -2x by subtracting 24 from both sides: -2x ≥ -24.
4. Divide both sides of the inequality by -2. Remember, when dividing or multiplying by a negative number, the inequality sign must be flipped: x ≤ 12.
5. The domain of the function is all x-values less than or equal to 12. In interval notation, the domain is (-∞, 12].

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 real-valued functions, this often involves identifying restrictions based on the operations involved, such as avoiding division by zero or ensuring that expressions under a square root are non-negative.

Square Root Function

A square root function, denoted as √(x), is defined only for non-negative values of x. This means that the expression inside the square root must be greater than or equal to zero to yield real number outputs. Understanding this property is crucial when determining the domain of functions that include square roots.

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 the context of finding the domain of the function f(x) = √(24 - 2x), solving the inequality 24 - 2x ≥ 0 will help identify the valid x-values that keep the expression under the square root non-negative.