Find the domain of each rational expression. (x + 3) / (x - 6)

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Identify the rational expression given: \(\frac{x + 3}{x - 6}\).

Recall that the domain of a rational expression includes all real numbers except those that make the denominator equal to zero.

Set the denominator equal to zero to find the values to exclude: \(x - 6 = 0\).

Solve the equation \(x - 6 = 0\) to find \(x = 6\).

Conclude that the domain is all real numbers except \(x = 6\), because at \(x = 6\) the denominator becomes zero and the expression is undefined.

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주요 개념

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Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding rational expressions involves knowing how to simplify, evaluate, and analyze them, especially focusing on restrictions caused by the denominator.

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 rational expressions, the domain excludes any values that make the denominator zero, as division by zero is undefined.

Finding Restrictions on the Domain

To find the domain of a rational expression, identify values that make the denominator zero and exclude them. For example, setting the denominator equal to zero and solving for x reveals these restricted values.