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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 64
Chapter 4, Problem 64

Find the domain of each function. f(x)=x2x−1−1f(x) = \(\sqrt{\frac{x}{2x - 1}\) - 1}

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

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For functions involving roots or fractions, the domain is restricted to values that keep the expression valid, such as avoiding division by zero or taking the square root of negative numbers.
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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.
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Rational Expressions and Denominator Restrictions

For rational expressions, the denominator cannot be zero because division by zero is undefined. When the function contains a fraction inside the square root, you must exclude values of x that make the denominator zero to find the valid domain.
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Related Practice
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Find the domain of each function. f(x)=2xx+1−1f(x) = \(\sqrt{\frac{2x}{x + 1}\) - 1}