Find the domain of each rational expression. (2x - 4) / (x + 7)

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Identify the rational expression given: \(\frac{2x - 4}{x + 7}\).

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: \(x + 7 = 0\).

Solve the equation for \(x\): \(x = -7\).

Conclude that the domain is all real numbers except \(x = -7\), which can be written in interval notation as \((-\infty, -7) \cup (-7, \infty)\).

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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, set the denominator equal to zero and solve for x. The solutions are excluded from the domain because they cause division by zero. The domain includes all real numbers except these restricted values.