Worksheets · Calculus

Piecewise functions worksheet for Calculus

Calculus uses piecewise functions to set up continuity and limits at the break points, so the first job is reading which rule owns each input. Expect three-piece rules with squares, cubes and absolute values, and graphs whose rule you write from the pieces.

Also taught in Precalculus, Algebra 1 and Algebra 2, each at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Piecewise Functions

Date Period

Evaluate each function, or answer as asked.

  1. Evaluate each piecewise function

    Find f(−1) for f(x)={x3+4x<−4−8−4≤x<−1−3 ⁣∣x+1∣+2x≥−1
  2. Find f(1) and f(3) for f(x)={−2 ⁣∣x∣+1x<1x3+31≤x<3−2x+5x≥3
  3. Find f(f(−3)) for f(x)={x2−6x≤0x3−60<x≤4∣x−1∣+6x>4
  4. Find f(−4) and f(0) for f(x)={2x3+8x<−4−x2−x+1−4≤x<0−2 ⁣∣x∣+1x≥0
  5. Find f(2) for f(x)={−x+6x<−1−2 ⁣∣x−3∣−2−1≤x<22x3−1x≥2
  6. Find f(f(−2)) for f(x)={−x3−3x≤−45−4<x≤−2−3 ⁣∣x−1∣+5x>−2
  7. Find f(−4) and f(−1) for f(x)={−3 ⁣∣x∣−1x≤−42−4<x≤−12x3+4x>−1

  1. Write the rule from the graph, then use it

    Write the three-piece rule for the graphed function f. Then use it to find f(−16) and f(24).−10−10−5−5551010
  2. Write the three-piece rule for the graphed function f. Then use it to find f(−30) and f(14).−10−10−5−5551010
  3. Write the three-piece rule for the graphed function f. Then use it to find f(−24) and f(11).−10−10−5551010−5
  4. Write the three-piece rule for the graphed function f. Then use it to find f(−14) and f(14).−10−10−5−5101055
  5. Write the three-piece rule for the graphed function f. Then use it to find f(−12) and f(20).−10−10−5−5551010
  6. Write the three-piece rule for the graphed function f. Then use it to find f(−18) and f(28).−10−10−5−5551010
  7. Write the three-piece rule for the graphed function f. Then use it to find f(−27) and f(12).−10−10−5551010−5

  1. Solve each equation, checking each piece’s condition

    Solve f(x)=4, where f(x)={∣x+1∣+3x≤040<x≤4−x3+5x>4
  2. Solve f(x)=−2, where f(x)={3x+7x≤−2−x3+6−2<x≤32 ⁣∣x−5∣−4x>3
  3. Solve f(x)=0, where f(x)={−2 ⁣∣x+3∣+10x≤−4x3−4<x≤−10x>−1
  4. Solve f(x)=3, where f(x)={∣x−2∣−2x<−14x−9−1≤x<2−x3+11x≥2
  5. Solve f(x)=−6, where f(x)={−2 ⁣∣x−1∣−4x<−3x3−7−3≤x<2−6x≥2
  6. Solve f(x)=−4, where f(x)={−x3−12x<02 ⁣∣x+1∣−100≤x<4−x+2x≥4

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

Sigma Prep · sigmaprep.io/worksheets

Name

Piecewise FunctionsAnswer keyVersion 3

Date Period

Evaluate each function, or answer as asked.

  1. Evaluate each piecewise function

    Find f(−1) for f(x)={x3+4x<−4−8−4≤x<−1−3 ⁣∣x+1∣+2x≥−12
  2. Find f(1) and f(3) for f(x)={−2 ⁣∣x∣+1x<1x3+31≤x<3−2x+5x≥3f(1)=4, f(3)=−1
  3. Find f(f(−3)) for f(x)={x2−6x≤0x3−60<x≤4∣x−1∣+6x>421
  4. Find f(−4) and f(0) for f(x)={2x3+8x<−4−x2−x+1−4≤x<0−2 ⁣∣x∣+1x≥0f(−4)=−11, f(0)=1
  5. Find f(2) for f(x)={−x+6x<−1−2 ⁣∣x−3∣−2−1≤x<22x3−1x≥215
  6. Find f(f(−2)) for f(x)={−x3−3x≤−45−4<x≤−2−3 ⁣∣x−1∣+5x>−2−7
  7. Find f(−4) and f(−1) for f(x)={−3 ⁣∣x∣−1x≤−42−4<x≤−12x3+4x>−1f(−4)=−13, f(−1)=2

  1. Write the rule from the graph, then use it

    Write the three-piece rule for the graphed function f. Then use it to find f(−16) and f(24).−10−10−5−5551010f(x)={32x+4x≤−42−4<x≤612x−1x>6 (the pieces meet at x=6, so either may include it); f(−16)=−20, f(24)=11
  2. Write the three-piece rule for the graphed function f. Then use it to find f(−30) and f(14).−10−10−5−5551010f(x)={−2x−3x≤−48−4<x<4−32x+11x≥4; f(−30)=57, f(14)=−10
  3. Write the three-piece rule for the graphed function f. Then use it to find f(−24) and f(11).−10−10−5551010−5f(x)={−3x−16x<−4−12x−4−4≤x≤4−2x+2x>4 (the pieces meet at x=4, so either may include it); f(−24)=56, f(11)=−20
  4. Write the three-piece rule for the graphed function f. Then use it to find f(−14) and f(14).−10−10−5−5101055f(x)={12x+7x<−4−x+4−4≤x<6−2x+14x≥6; f(−14)=0, f(14)=−14
  5. Write the three-piece rule for the graphed function f. Then use it to find f(−12) and f(20).−10−10−5−5551010f(x)={−x−9x≤−612x−6<x≤22x−6x>2 (the pieces meet at x=−6, so either may include it); f(−12)=3, f(20)=34
  6. Write the three-piece rule for the graphed function f. Then use it to find f(−18) and f(28).−10−10−5−5551010f(x)={32x+3x≤−4−3−4<x<22x−2x≥2 (the pieces meet at x=−4, so either may include it); f(−18)=−24, f(28)=54
  7. Write the three-piece rule for the graphed function f. Then use it to find f(−27) and f(12).−10−10−5551010−5f(x)={2x+8x<−2−12x+6−2≤x<632x−1x≥6; f(−27)=−46, f(12)=17

  1. Solve each equation, checking each piece’s condition

    Solve f(x)=4, where f(x)={∣x+1∣+3x≤040<x≤4−x3+5x>4x=−2 or x=0 or 0<x≤4
  2. Solve f(x)=−2, where f(x)={3x+7x≤−2−x3+6−2<x≤32 ⁣∣x−5∣−4x>3x=−3 or x=2 or x=4 or x=6
  3. Solve f(x)=0, where f(x)={−2 ⁣∣x+3∣+10x≤−4x3−4<x≤−10x>−1x=−8 or x>−1
  4. Solve f(x)=3, where f(x)={∣x−2∣−2x<−14x−9−1≤x<2−x3+11x≥2x=−3 or x=2
  5. Solve f(x)=−6, where f(x)={−2 ⁣∣x−1∣−4x<−3x3−7−3≤x<2−6x≥2x=1 or x≥2
  6. Solve f(x)=−4, where f(x)={−x3−12x<02 ⁣∣x+1∣−100≤x<4−x+2x≥4x=−2 or x=2 or x=6

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How to do these

f(−2) for f(x)={x2+1x<−23∣x∣−4−2≤x<12xx≥1

  1. -2 is not less than -2, so the first rule is out.
  2. -2 ≤ -2 < 1, so use 3|x| - 4.
  3. 3(2) - 4 = 2.

The answer is 2.

Where students go wrong

Using the wrong rule at a break point. -2 ≤ x includes -2 and x < -2 does not, so at x = -2 only the middle rule applies, however close the other one comes.

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

What you can put on this worksheet

Evaluate a piecewise function
Find f(−1) and f(6) for f(x)={xx<13x≥1 → f(−1)=−1, f(6)=3
From a graph: read values, or write the rule
Use the graph to find f(−7), f(−6) and f(−5). → f(−7)=−3, f(−6)=−5, f(−5)=−2
Solve f(x) = k
Solve f(x)=4, where f(x)={2x+12x<17x≥1 → x=−4

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 3: evaluate a piecewise function, from a graph: read values, or write the rule and solve f(x) = k. 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.