In Exercises 1–30, find the domain of each function. g(x) = √(x −2)/(x-5)

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1. Identify the conditions for the square root function: The expression inside the square root, \(x - 2\), must be greater than or equal to zero. This gives us the inequality \(x - 2 \geq 0\).
2. Solve the inequality \(x - 2 \geq 0\) to find the values of \(x\) that satisfy this condition. This simplifies to \(x \geq 2\).
3. Identify the condition for the denominator: The expression in the denominator, \(x - 5\), must not be equal to zero to avoid division by zero. This gives us the inequality \(x - 5 \neq 0\).
4. Solve the inequality \(x - 5 \neq 0\) to find the values of \(x\) that satisfy this condition. This simplifies to \(x \neq 5\).
5. Combine the conditions from steps 2 and 4 to find the domain of the function. The domain is all \(x\) such that \(x \geq 2\) and \(x \neq 5\).

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 functions, the domain excludes values that make the denominator zero, while for radical functions, it excludes values that result in taking the square root of a negative number.

Square Root Function

A square root function, such as √(x - 2), is defined only for non-negative values of the expression inside the square root. This means that the expression must be greater than or equal to zero, which leads to the condition x - 2 ≥ 0, or x ≥ 2, to ensure the output is real.

Rational Function

A rational function is a function that can be expressed as the ratio of two polynomials. In the case of g(x) = √(x - 2)/(x - 5), the denominator (x - 5) cannot be zero, which means x cannot equal 5. This restriction must be considered when determining the overall domain of the function.