In Exercises 1–6, find all numbers that must be excluded from the domain of each rational expression. 7/(x−3)

Verified step by step guidance

1. Identify the denominator of the rational expression, which is the expression located below the division line. In this case, the denominator is (x−3).
2. Understand that the domain of a rational expression includes all real numbers except those that make the denominator equal to zero, as division by zero is undefined.
3. Set the denominator equal to zero and solve for x. This means solving the equation x−3 = 0.
4. Add 3 to both sides of the equation to isolate x, resulting in x = 3.
5. Conclude that the number that must be excluded from the domain of the rational expression 7/(x−3) is x = 3.

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 they can have restrictions on their domain, specifically where the denominator equals zero, as division by zero is 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 make the denominator zero, as these values would lead to undefined expressions.

Finding Excluded Values

To find the values that must be excluded from the domain of a rational expression, one must set the denominator equal to zero and solve for x. In the expression 7/(x−3), setting x−3=0 reveals that x=3 must be excluded from the domain, as it would make the denominator zero.