Worksheets · Calculus

Calculus piecewise functions worksheet

A piecewise function is several rules with a note about when each one applies. The only new step is reading the condition first and picking the right rule, which is exactly what the break point tests.

Select your difficulty

Pick more than one for a sheet that mixes them.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Piecewise Functions

Date Period

Evaluate each function, or find the value asked for.

  1. Evaluate f(−12) for f(x)={−4x2+4x<−43x−4≤x<1−2x2−7x≥1
  2. Evaluate f(−3) for f(x)={−4∣x∣+1x<−4−5∣x∣−1−4≤x<12x+5x≥1
  3. Evaluate f(−2) for f(x)={2x3+9x<−2−2∣x∣+3−2≤x<05∣x∣+8x≥0
  4. Evaluate f(−2) for f(x)={∣x∣+3x<−6−6x2−6−6≤x<−2−3x3−3x≥−2
  5. Evaluate f(3) for f(x)={−4x2+1x<−2−x2−8−2≤x<34x3−7x≥3
  6. Evaluate f(72) for f(x)={−x2+9x<66x2−86≤x<115x+6x≥11
  7. Evaluate f(−5) for f(x)={2x2−2x<−6−2∣x∣+8−6≤x<−1−2x−8x≥−1
  8. Evaluate f(6) for f(x)={−2∣x∣+1x<4x3−64≤x<7x2−3x≥7
  9. Evaluate f(32) for f(x)={−x+6x<42x2+34≤x<86x−8x≥8
  10. Evaluate f(−1) for f(x)={−6x2−7x<−14x3−2−1≤x<4−2x3+9x≥4
  11. Evaluate f(3) for f(x)={−x2−9x<−14x3+6−1≤x<36∣x∣−9x≥3
  12. Evaluate f(92) for f(x)={3x3−9x<35x2+73≤x<5x3−2x≥5
  13. Evaluate f(−5) for f(x)={6x2+2x<−53x−6−5≤x<04x3−9x≥0
  14. Evaluate f(5) for f(x)={4xx<−1∣x∣+5−1≤x<2−4x3+7x≥2
  15. Evaluate f(−32) for f(x)={−6∣x∣−2x<−54x−1−5≤x<−14x+5x≥−1
  16. Evaluate f(−3) for f(x)={2x3−2x<−2−4x3+6−2≤x<0−∣x∣−2x≥0
  17. Evaluate f(232) for f(x)={x−1x<4−4∣x∣−14≤x<9−5x2+8x≥9
  18. Evaluate f(3) for f(x)={−4x3−9x<3−4x−53≤x<5−2x3−4x≥5
  19. Evaluate f(2) for f(x)={−x3−3x<43x3+24≤x<6−3x3−8x≥6
  20. Evaluate f(72) for f(x)={−∣x∣−5x<6−3∣x∣+66≤x<9−5x3+6x≥9

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Piecewise FunctionsAnswer keyVersion 1

Date Period

Evaluate each function, or find the value asked for.

  1. Evaluate f(−12) for f(x)={−4x2+4x<−43x−4≤x<1−2x2−7x≥1−32
  2. Evaluate f(−3) for f(x)={−4∣x∣+1x<−4−5∣x∣−1−4≤x<12x+5x≥1−16
  3. Evaluate f(−2) for f(x)={2x3+9x<−2−2∣x∣+3−2≤x<05∣x∣+8x≥0−1
  4. Evaluate f(−2) for f(x)={∣x∣+3x<−6−6x2−6−6≤x<−2−3x3−3x≥−221
  5. Evaluate f(3) for f(x)={−4x2+1x<−2−x2−8−2≤x<34x3−7x≥3101
  6. Evaluate f(72) for f(x)={−x2+9x<66x2−86≤x<115x+6x≥11−134
  7. Evaluate f(−5) for f(x)={2x2−2x<−6−2∣x∣+8−6≤x<−1−2x−8x≥−1−2
  8. Evaluate f(6) for f(x)={−2∣x∣+1x<4x3−64≤x<7x2−3x≥7210
  9. Evaluate f(32) for f(x)={−x+6x<42x2+34≤x<86x−8x≥892
  10. Evaluate f(−1) for f(x)={−6x2−7x<−14x3−2−1≤x<4−2x3+9x≥4−6
  11. Evaluate f(3) for f(x)={−x2−9x<−14x3+6−1≤x<36∣x∣−9x≥39
  12. Evaluate f(92) for f(x)={3x3−9x<35x2+73≤x<5x3−2x≥54334
  13. Evaluate f(−5) for f(x)={6x2+2x<−53x−6−5≤x<04x3−9x≥0−21
  14. Evaluate f(5) for f(x)={4xx<−1∣x∣+5−1≤x<2−4x3+7x≥2−493
  15. Evaluate f(−32) for f(x)={−6∣x∣−2x<−54x−1−5≤x<−14x+5x≥−1−7
  16. Evaluate f(−3) for f(x)={2x3−2x<−2−4x3+6−2≤x<0−∣x∣−2x≥0−56
  17. Evaluate f(232) for f(x)={x−1x<4−4∣x∣−14≤x<9−5x2+8x≥9−26134
  18. Evaluate f(3) for f(x)={−4x3−9x<3−4x−53≤x<5−2x3−4x≥5−17
  19. Evaluate f(2) for f(x)={−x3−3x<43x3+24≤x<6−3x3−8x≥6−11
  20. Evaluate f(72) for f(x)={−∣x∣−5x<6−3∣x∣+66≤x<9−5x3+6x≥9−172

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

f(−2) for f(x)={2x+1x<0x2−3x≥0

  1. Minus 2 is less than 0.
  2. So the first rule applies.
  3. Two times minus 2 plus 1.

The answer is −3.

Where students go wrong

Using the wrong branch at the break point. Look at whether the condition says less than or less than or equal to; that single line decides which rule owns the boundary, and it is the input the question will pick.

Also called evaluating piecewise functions, piecewise defined functions or continuity.

What you can put on this worksheet

Evaluating a piecewise function
Evaluate f(−1) for f(x)={x+9x≤32x>3 → 8
The value that makes it continuous
Find k so that f is continuous: f(x)={kx−4x≤20x>2 → 2

Questions about these worksheets

Yes. Every piecewise functions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the piecewise functions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 2: evaluating a piecewise function and the value that makes it continuous. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the wrong branch at the break point. Look at whether the condition says less than or less than or equal to; that single line decides which rule owns the boundary, and it is the input the question will pick.