51.7 Elementary Multivariable Restrictions

Elementary Multivariable Restrictions are the domain restrictions that arise when a rational expression's denominator involves more than one variable, extending the single-variable exclusion technique to account for combinations of variable values, rather than isolated numbers, that make the denominator equal zero. Because a multivariable denominator can equal zero along an entire curve or relationship between the variables rather than at a single point, these restrictions are typically stated as a condition relating the variables to one another, instead of a short list of excluded numbers.

The underlying principle, that the denominator must never equal zero, is unchanged from the single-variable case; only the form the restriction takes, a relationship between two or more variables rather than a finite list, differs.

Locating the Factors

Multivariable Denominator Factor Inventory

The denominator is factored just as in the single-variable case, and every factor appearing in that factorization is inventoried, whether that factor involves one variable alone or a combination of several variables together.

Conditions From Single-Variable Factors

Individual Variable Nonzero Conditions

When a factor of the denominator involves only one of the variables, the condition it produces is a simple exclusion on that single variable, exactly as in the single-variable case, independent of whatever value the other variables take.

Conditions From Multivariable Factors

Linear Multivariable Denominator Condition

When a factor of the denominator involves more than one variable, such as y − 1 involving only y in this example, or an expression like x + y involving both, the resulting condition excludes every combination of variable values that makes that specific factor equal to zero, which is generally a relationship rather than a single number.

Combining Every Condition

Combined Domain Condition Statement

Every individual condition produced by every factor of the denominator is combined together, using "and," into a single statement describing the full set of variable combinations for which the multivariable rational expression is defined.

Simplification in the Multivariable Case

Multivariable Common Factor Cancellation

Common-factor cancellation applies to multivariable rational expressions the same way it applies to single-variable ones: any factor, whether single-variable or multivariable, that appears in both the fully factored numerator and denominator can be cancelled once.

Original Multivariable Restriction Retention

Exactly as in the single-variable case, every domain condition determined from the original, unsimplified denominator must still be carried forward and stated alongside the simplified multivariable expression, even for factors that have since been cancelled away.

Evaluating a Multivariable Expression

Allowed Ordered-Value Evaluation

Evaluating a multivariable rational expression at a specific point requires substituting a value for every variable simultaneously, then checking that this entire combination of values satisfies every domain condition before computing a numerical result; a combination that violates even one condition means the expression is undefined there, regardless of whether it would satisfy the other conditions individually.