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 \(x - 3\).
2. Recall that the domain of a rational expression excludes any values that make the denominator equal to zero, because division by zero is undefined.
3. Set the denominator equal to zero to find the excluded values: \(x - 3 = 0\).
4. Solve the equation for \(x\): add 3 to both sides to get \(x = 3\).
5. Conclude that the number \(x = 3\) must be excluded from the domain of the expression.

Key Concepts

Domain of a Rational Expression

The domain of a rational expression includes all real numbers except those that make the denominator zero. Since division by zero is undefined, these values must be excluded to ensure the expression is valid.

Identifying Values That Make the Denominator Zero

To find excluded values, set the denominator equal to zero and solve for the variable. These solutions are the numbers that cannot be in the domain because they cause division by zero.

Simplifying Rational Expressions

Understanding how to simplify rational expressions helps in identifying restrictions on the domain. Even after simplification, any value that originally made the denominator zero remains excluded from the domain.