In Exercises 61–64, find the domain of each function.f(x) = √(x/(2x - 1) - 1)

Verified step by step guidance

1. Identify the inner function and set its domain. The inner function here is the expression under the square root, which is \(x/(2x - 1) - 1\).
2. Simplify the expression inside the square root to find where it is defined. The expression simplifies to \((x - (2x - 1))/(2x - 1)\).
3. Set the denominator of the simplified expression not equal to zero to avoid division by zero. Solve \(2x - 1 \neq 0\).
4. Ensure the expression under the square root is non-negative because the square root of a negative number is not defined in the set of real numbers. Set \((x - (2x - 1))/(2x - 1) \geq 0\) and solve the inequality.
5. Combine the conditions from the previous steps to find the overall domain of the function, considering both the non-zero denominator and the non-negative requirement under the square root.

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, while for rational functions, the denominator cannot be zero. Understanding the domain is crucial for determining valid inputs that yield real outputs.

Square Root Function

A square root function is defined as f(x) = √g(x), where g(x) must be greater than or equal to zero for the function to yield real numbers. In this case, the expression inside the square root, x/(2x - 1) - 1, must be non-negative. This condition leads to inequalities that must be solved to find the domain.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. In the function f(x) = √(x/(2x - 1) - 1), the term 2x - 1 in the denominator must not equal zero, as this would make the expression undefined. Identifying values that make the denominator zero is essential for determining the domain of the function.