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Grade 12 · Math · Calculus

12th grade Piecewise continuity and the Intermediate Value Theorem worksheet

6 printable versionsBuilding Confidence10 questions · answer keyFree · no account

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Piecewise continuity and the Intermediate Value Theorem

Grade 12 Math Worksheet - Building Confidence

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  1. f(x) = 3x + 1 for x < 2 and f(x) = x + 5 for x ≥ 2. Ari checks the join. What happens at x = 2?

    • a jump at x = 2
    • a hole at x = 2
    • continuous at x = 2

  2. f(x) = x + 2 for x < 3 and f(x) = 2x − 1 for x ≥ 3. Esme checks the join. What happens at x = 3?

    • continuous at x = 3
    • a jump at x = 3
    • a hole at x = 3

  3. f(x) = 4x + 6 for x < -1 and f(x) = -x + 3 for x ≥ -1. Jun checks the join. What happens at x = -1?

    • a jump at x = -1
    • continuous at x = -1
    • a hole at x = -1

  4. f(x) = 2x + 1 for x < 3 and f(x) = -x + 12 for x ≥ 3. Kira needs f(3). What is it?

  5. f(x) = 5x − 3 for x < 2 and f(x) = 3x − 2 for x ≥ 2. Luz needs f(2). What is it?

  6. f(x) = -2x + 8 for x < 1 and f(x) = 4x + 3 for x ≥ 1. Noah needs f(1). What is it?

  7. f is continuous on [0, 4], with f(0) = -3 and f(4) = 5. Pia asks about a zero. Which is true?

    • yes: f must cross 0 between 0 and 4
    • no zero is possible on [0, 4]
    • the IVT does not promise a zero on [0, 4]

  8. f is continuous on [1, 6], with f(1) = 2 and f(6) = 9. Ren asks about a zero. Which is true?

    • the IVT does not promise a zero on [1, 6]
    • no zero is possible on [1, 6]
    • yes: f must cross 0 between 1 and 6

  9. f is continuous on [2, 5], with f(2) = 7 and f(5) = -1. Tala asks about a zero. Which is true?

    • no zero is possible on [2, 5]
    • the IVT does not promise a zero on [2, 5]
    • yes: f must cross 0 between 2 and 5

  10. f is continuous on [0, 3], with f(0) = 4 and f(3) = 10. Zee asks about a zero. Which is true?

    • no zero is possible on [0, 3]
    • the IVT does not promise a zero on [0, 3]
    • yes: f must cross 0 between 0 and 3
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12th grade Piecewise continuity and the Intermediate Value Theorem worksheet – printable
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About this worksheet

Piecewise continuity and the Intermediate Value Theorem. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can check a piecewise function at its join and find f at the join.
  • I can choose constants that make a piecewise function continuous.
  • I can use the IVT to promise a zero or a value, and say when it promises nothing.

Standard: none (beyond CCSS)

Words to know

join: The x where a piecewise function changes rule.

Intermediate Value Theorem (IVT): A continuous f on [a, b] hits every height between f(a) and f(b).

sign change: Neighboring values on opposite sides of 0.

Common mistakes

Thought a change of rule always breaks the graph. If both pieces give the same height at the join, it is continuous.

Thought the join point is missing. One of the pieces always includes it (≥ or ≤).

Checked only one piece at the join. Both pieces must give the same height.

Questions parents ask

What is the most common mistake with piecewise continuity and the intermediate value theorem?

Thought a change of rule always breaks the graph. If both pieces give the same height at the join, it is continuous.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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