Find the domain of each function. f(x) = 1/√(x - 3)

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Identify the expression inside the square root in the denominator: \(\sqrt{\\(x - 3\")}\). Since the square root is in the denominator, the expression inside the root must be strictly greater than zero to avoid division by zero and undefined values.

Set up the inequality for the radicand (the expression inside the square root): \(x - 3 > 0\).

Solve the inequality for \(x\): add 3 to both sides to get \(x > 3\).

Interpret the solution: the domain of the function consists of all real numbers greater than 3, because for these values the denominator is defined and not zero.

Express the domain in interval notation: \((3, \infty)\).

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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 is the set of all possible input values (x-values) for which the function is defined. For rational and root functions, the domain is restricted by values that cause division by zero or taking the square root of a negative number.

Square Root Function Restrictions

A square root function requires the radicand (expression inside the root) to be greater than or equal to zero to produce real number outputs. For f(x) = 1/√(x - 3), the expression x - 3 must be strictly greater than zero because the root is in the denominator.

Division by Zero

Division by zero is undefined in mathematics. Since the square root is in the denominator, the radicand cannot be zero because that would make the denominator zero, causing the function to be undefined at that point.