In Exercises 1–30, find the domain of each function. f(x) = 1/√(x - 3)

Verified step by step guidance

1. Identify the expression under the square root: \( x - 3 \).
2. Recognize that the expression under the square root must be greater than zero for the function to be defined, since the square root of a negative number is not real and division by zero is undefined.
3. Set up the inequality: \( x - 3 > 0 \).
4. Solve the inequality for \( x \): Add 3 to both sides to get \( x > 3 \).
5. Conclude that the domain of the function \( f(x) = \frac{1}{\sqrt{x - 3}} \) is all real numbers greater than 3, which can be expressed in interval notation as \((3, \infty)\).

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 rational and radical functions, certain values may lead to undefined expressions, such as division by zero or taking the square root of a negative number. Understanding the domain is crucial for determining where the function can be evaluated.

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. In the context of the given function f(x) = 1/√(x - 3), the expression x - 3 must be positive to ensure that the square root is defined and yields a real number.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values, indicating whether one is greater than, less than, or equal to the other. To find the domain of the function f(x) = 1/√(x - 3), we set up the inequality x - 3 > 0, which simplifies to x > 3. Solving this inequality helps identify the valid input values for the function.