Find the domain of each function. f(x) = √(x - 3)

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Step 1: Recall that the domain of a function is the set of all possible input values (x-values) for which the function is defined.

Step 2: For the square root function √(x - 3), the expression inside the square root (x - 3) must be greater than or equal to 0 because the square root of a negative number is not defined in the set of real numbers.

Step 3: Set up the inequality x - 3 ≥ 0 to determine the values of x that make the function valid.

Step 4: Solve the inequality x - 3 ≥ 0 by adding 3 to both sides, resulting in x ≥ 3.

Step 5: Conclude that the domain of the function is all x-values such that x ≥ 3. In interval notation, this is written as [3, ∞).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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. Understanding the domain is crucial because it determines the values that can be substituted into the function without resulting in undefined expressions, such as division by zero or taking the square root of a negative number.

Square Root Function

A square root function, denoted as f(x) = √(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. For the function f(x) = √(x - 3), this condition leads to the requirement that x - 3 must be non-negative, which directly influences the domain.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values, indicating whether one is less than, greater than, or equal to the other. In finding the domain of the function f(x) = √(x - 3), we set up the inequality x - 3 ≥ 0, which helps us determine the minimum value of x that keeps the function defined, thus allowing us to find the domain.