Find the domain of each function. $f(x) = \sqrt{\frac{2x}{x + 1} - 1}$

Verified step by step guidance

1. Identify the function given: \(f(x) = \sqrt{\frac{2x}{x + 1} - 1}\).
2. Recall that the expression inside the square root must be greater than or equal to zero for the function to be defined. So, set up the inequality: \(\frac{2x}{x + 1} - 1 \geq 0\).
3. Combine the terms on the left side over a common denominator: \(\frac{2x}{x + 1} - \frac{x + 1}{x + 1} \geq 0\), which simplifies to \(\frac{2x - (x + 1)}{x + 1} \geq 0\).
4. Simplify the numerator: \(\frac{2x - x - 1}{x + 1} = \frac{x - 1}{x + 1} \geq 0\).
5. Determine the domain by finding where the rational expression \(\frac{x - 1}{x + 1}\) is greater than or equal to zero, and exclude values that make the denominator zero (i.e., \(x \neq -1\)).

Key Concepts

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. For functions involving roots or fractions, certain values may be excluded to avoid undefined expressions like division by zero or taking the square root of a negative number.

Square Root Function Restrictions

When a function includes a square root, the expression inside the root (the radicand) must be greater than or equal to zero. This ensures the output is a real number, as square roots of negative numbers are not defined in the set of real numbers.

Rational Expressions and Denominator Restrictions

For rational expressions, the denominator cannot be zero because division by zero is undefined. Identifying values of x that make the denominator zero is essential to exclude them from the domain.