excluded value

An excluded value is an input that you are simply not allowed to plug in, because doing so would make the denominator zero — and dividing by zero is meaningless. Think of it as a “No Entry” sign posted on certain numbers along the number line.

To find the excluded values of a rational expression, set its denominator equal to zero and solve. Each solution is an excluded value. For 1/(x - 5) the denominator vanishes at x = 5, so x = 5 is excluded. For x/((x - 1)(x + 3)) both x = 1 and x = -3 are excluded.

Excluded values must be tracked even after you simplify. If a factor cancels, its excluded value does not come back to life — the original expression was never defined there, so the simplified one inherits the same gap. Keeping the full list is what lets you state the true domain of the expression.