Piecewise Functions • Topic 3 of 3

Evaluations

To evaluate a piecewise function at an input, first find which interval the input belongs to, then substitute into that piece only. Consider f(x) = 2x + 1 for x < 0 and f(x) = x² for x ≥ 0. Then f(−3) uses the first piece (since −3 < 0): 2(−3) + 1 = −5, while f(4) uses the second (since 4 ≥ 0): 4² = 16. The most common error is using the wrong branch, so always check the inequality before substituting — especially at boundary values, where the “equal to” sign decides which piece applies. The SAT tests exactly this two-step process: select the branch, then evaluate.

Evaluating a piecewise function: a dashed vertical line rises from an x-value to the green branch, then a dashed horizontal line runs left to the y-axis marking the output.Evaluating a piecewise functionOx = af(a)Pick the branch for the input, then read the output

✅ Solved examples

1. f(x) = 2x + 1 for x < 0, x² for x ≥ 0. Find f(4).
4 ≥ 0, so use x²: 4² = 16.
2. Same f. Find f(−3).
−3 < 0, so use 2x + 1: 2(−3) + 1 = −5.
3. f(x) = 5x − 2 for x < 2, x² for x ≥ 2. Find f(3).
3 ≥ 2, so x²: 9.
4. Same f. Find f(1).
1 < 2, so 5x − 2: 5(1) − 2 = 3.

✏️ Practice — try these, take hints as needed

1. f(x) = 3x for x < 1, x² for x ≥ 1. Find f(4).
4 ≥ 1.
Use x².
4².
16.
2. Same f. Find f(0).
0 < 1.
Use 3x.
—
0.
3. f(x) = 2x + 1 for x < 2, x² for x ≥ 2. Find f(5).
5 ≥ 2.
5².
—
25.
4. Same f. Find f(1).
1 < 2.
2(1) + 1.
—
3.
5. f(x) = 4x − 3 for x < 0, x² for x ≥ 0. Find f(−2).
−2 < 0.
4(−2) − 3.
—
−11.

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Graphs Take the chapter test →