Piecewise Functions • Topic 1 of 3

Definition

A piecewise function is defined by two or more rules, each used on a specified interval of the domain. To work with one, first decide which interval the input belongs to, then use only that interval’s rule. For a function defined as one rule for x < 3 and another for x ≥ 3, an input of 2 uses the first rule and an input of 5 uses the second; an input of exactly 3 uses whichever piece includes the “equal to” boundary. The absolute-value function is itself a two-piece example. The SAT mainly tests choosing the correct branch for a given input, so check the inequality boundaries carefully.

A piecewise function: a blue line for x less than 0 ending in an open circle at the origin, and a green curve for x greater than or equal to 0 starting with a filled circle.A piecewise functionODifferent rule on each interval of x

✅ Solved examples

1. A function uses one rule for x < 3 and another for x ≥ 3. Which rule applies at x = 5?
Since 5 ≥ 3, the second rule.
2. For the same function, which rule applies at x = 1?
Since 1 < 3, the first rule.
3. How many pieces define |x| as a piecewise function?
Two (x ≥ 0 and x < 0).
4. A rule for x ≤ 0 and another for x > 0. Which applies at x = 0?
The first (x ≤ 0 includes 0).

✏️ Practice — try these, take hints as needed

1. A function uses one rule for x < 2 and another for x ≥ 2. Which applies at x = 4?
Is 4 < 2 or ≥ 2?
4 ≥ 2.
—
The second rule.
2. For the same function, which applies at x = 0?
0 < 2.
—
—
The first rule.
3. A rule for x < 5 and another for x ≥ 5. Which applies at x = 5?
Which piece has “equal to”?
x ≥ 5.
—
The second rule.
4. A rule for x ≤ −1 and another for x > −1. Which applies at x = 3?
3 > −1.
—
—
The second rule.
5. How many rules make up a piecewise function, at minimum?
“Pieces”.
—
—
Two.

📝 Topic test — 8 questions

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