Concept explainerCalculus I · Algebra IIIntermediate

Domain restrictions in rational expressions

A rational expression has no value wherever its original denominator is zero. Simplifying the formula does not restore an excluded input.

LaTeX article Updated July 13, 2026

\frac{x^2-9}{x-3}=x+3,\quad x\ne3

Division by zero is not defined

A denominator records a division. Inputs that make it zero are outside the expression's domain, even when the numerator is also zero.

Factor complicated denominators and set each factor equal to zero to identify all exclusions.

\frac1{(x-2)(x+5)}:\quad x\ne2,-5

A canceled factor leaves a hole

Canceling a common nonzero factor produces a simpler expression with the same values everywhere the original was defined. At the canceled zero, the original still has no value.

Graphically, that missing input appears as a removable discontinuity or hole rather than a vertical asymptote.

\frac{(x-3)(x+3)}{x-3}=x+3,\ x\ne3

Restrictions travel through algebra

When solving an equation or combining expressions, record restrictions at the start and compare every candidate solution with them at the end.

A candidate that violates an original denominator is extraneous, even if it satisfies a cleared equation.

x\notin\{\text{denominator zeros}\}

Worked example

Common mistakes

Finding restrictions after cancellation.
Treating 0/0 as zero.
Reporting only the restriction still visible in the simplified denominator.

Keep these ideas

Use the original denominator.
Cancellation does not restore excluded inputs.
Check solutions against restrictions.