Find the domain of each rational expression. See Example 1. x + 3 x - 6

Verified step by step guidance

1. Identify the rational expression: \( \frac{x + 3}{x - 6} \).
2. Recall that the domain of a rational expression is all real numbers except where the denominator is zero.
3. Set the denominator equal to zero to find the values that are not in the domain: \( x - 6 = 0 \).
4. Solve the equation \( x - 6 = 0 \) to find the value of \( x \) that makes the denominator zero.
5. Exclude the value found in the previous step from the domain of the rational expression.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial because their behavior, particularly regarding domain, is influenced by the values that make the denominator zero, which would make the expression undefined.

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 expressions, the domain excludes any values that cause the denominator to equal zero, as these values would lead to undefined expressions.

Finding Restrictions

To determine the domain of a rational expression, one must identify the values that make the denominator zero. This involves solving the equation formed by setting the denominator equal to zero and excluding these solutions from the domain, ensuring that the expression remains valid.