Piecewise Domains as Unions of Intervals

A student writes f(x) = { x+1 for x<0; x² for x≥2 } and then guesses the domain is "all real numbers except 0 to 2." That guess is wrong in two ways. It fails to say whether 0 or 2 are included, and it implies that the gap might be part of the domain if read carelessly. Piecewise domains as unions of intervals solve both problems by making every excluded gap explicit.

Union notation uses the symbol ∪ to join separate intervals. The domain for that function is (−∞, 0) ∪ [2, ∞). The parentheses around 0 mean x = 0 is excluded; the bracket at 2 means x = 2 is included. The gap [0, 2) is visually absent from the union, so no reader can misinterpret it as included. A comma or the word "or" between the two intervals is not standard interval notation and risks the same confusion you started with.

Work one example from the fact sheet to see the rule in action. Take f(x) = { 2x for -3 ≤ x < 1; 5 for x = 1; x − 3 for x > 1 }.

Identify Each Piece's Interval

The first piece runs from -3 to 1, including -3 but excluding 1. Write that as [-3, 1). The second piece covers only x = 1. Write that as [1, 1] or just include the point in the union. The third piece runs from 1 upward, excluding 1. Write that as (1, ∞).

Combine With Union

Join them: [-3, 1) ∪ [1, 1] ∪ (1, ∞). The point x = 1 appears in two intervals, but the union collapses them into [-3, 1] ∪ (1, ∞). The bracket at 1 in the first interval and the separate [1, 1] both show that x = 1 is included. The gap is nothing, the intervals meet at 1. If the second piece had been defined for x < 1 instead of x = 1, the domain would be [-3, 1) ∪ (1, ∞), and the gap would be exactly the single point x = 1.

When the pieces leave a real gap, the union notation must explicitly skip it. Suppose a function has one piece for x ≤ -2 and another for x ≥ 3. The domain is (-∞, -2] ∪ [3, ∞). The gap (−2, 3) is not written anywhere, so it is clearly excluded.

To verify your result, you can use an interval notation calculator to check that every x-value in your union produces a valid output and that no x-value in the gap does. The calculator will also flag a missing union symbol or an endpoint bracket that contradicts the piece definition.

Failure Case: Three or More Exclusions

A rational function with vertical asymptotes at x = -1, x = 2, and x = 5 produces domain (−∞, -1) ∪ (-1, 2) ∪ (2, 5) ∪ (5, ∞). Each excluded point creates its own gap. Writing "x ≠ -1, 2, 5" without separating the intervals is not interval notation. The union of four intervals is the only correct representation.

The key insight: union notation is not optional decoration. It is the exact representation needed for the domain result. Without it, a reader has to guess whether a comma means "and" or "or" and whether the gap is truly empty.

What the Union Symbol Does

The symbol ∪ tells the reader that the set is the combination of two or more separate intervals. Each interval is independent. There is no implication that values between the intervals are included, because the union symbol does not connect the intervals into one continuous range.

What Happens Without the Union

Writing "x < 0, x ≥ 2" is ambiguous. A student new to piecewise domains could read it as a single condition meaning x is less than 0 and also greater than or equal to 2, which is impossible. The union symbol removes that ambiguity.

This method works for any number of pieces and any number of gaps. The rule never changes: identify each piece's valid x-interval, write each in proper notation, and join them with ∪. That is the complete answer to how disconnected valid inputs should be written without implying that excluded gaps are included.