Worksheet

Piecewise continuity and the Intermediate Value Theorem

Grade 12 Math Worksheet - Challenge Me

NameDate

Circle the best answer, or write your answer on the line.

  1. f(x) = x² for x < 2, ax + b for 2 ≤ x < 4, and x² for x ≥ 4. Bo makes f continuous. As (a, b), what are a and b?

  2. f(x) = x² for x < -1, ax + b for -1 ≤ x < 2, and x² for x ≥ 2. Esme makes f continuous. As (a, b), what are a and b?

  3. f(x) = x² for x < 0, ax + b for 0 ≤ x < 3, and x² for x ≥ 3. Jun makes f continuous. As (a, b), what are a and b?

  4. f(0) = 5; f(1) = 8; f(2) = -2; f(3) = -6; f(4) = 3. Luz studies the table shown. What does the IVT say about zeros of f between x = 0 and x = 4?

    • no zero is promised
    • exactly 2 zeros
    • at least 2 zeros

  5. f(0) = -7; f(1) = 4; f(2) = 6; f(3) = 9; f(4) = 12. Milo studies the table shown. What does the IVT say about zeros of f between x = 0 and x = 4?

    • at least 1 zero
    • exactly 1 zero
    • no zero is promised

  6. f(0) = -8; f(1) = 3; f(2) = -5; f(3) = 7; f(4) = -9. Noah studies the table shown. What does the IVT say about zeros of f between x = 0 and x = 4?

    • exactly 4 zeros
    • at least 3 zeros
    • at least 4 zeros

  7. The air is 12 °C at 6 a.m. and 25 °C at noon, changing continuously. Pia asks whether it is ever exactly 20 °C. Which is true?

    • yes, if it changes at a steady rate
    • not promised: 20 °C is outside the range
    • yes, at some moment it equals 20 °C

  8. A drone climbs continuously from 20 m to 80 m in 10 seconds. Ren asks whether it is ever exactly 95 m. Which is true?

    • yes, at some moment it equals 95 m
    • yes, if it changes fast enough
    • not promised: 95 m is outside the range

  9. Soup cools continuously from 90 °C to 40 °C in an hour. Tala asks whether it is ever exactly 60 °C. Which is true?

    • yes, if it changes at a steady rate
    • not promised: 60 °C is outside the range
    • yes, at some moment it equals 60 °C

  10. A pool's depth drains continuously from 5 ft to 2 ft in a day. Zee asks whether it is ever exactly 1 ft. Which is true?

    • yes, at some moment it equals 1 ft
    • not promised: 1 ft is outside the range
    • yes, if it changes fast enough
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Piecewise continuity and the Intermediate Value Theorem

Answer Key - Grade 12 Math Worksheet - Challenge Me

Circle the best answer, or write your answer on the line.

  1. f(x) = x² for x < 2, ax + b for 2 ≤ x < 4, and x² for x ≥ 4. Bo makes f continuous. As (a, b), what are a and b?

    (6, -8)

  2. f(x) = x² for x < -1, ax + b for -1 ≤ x < 2, and x² for x ≥ 2. Esme makes f continuous. As (a, b), what are a and b?

    (1, 2)

  3. f(x) = x² for x < 0, ax + b for 0 ≤ x < 3, and x² for x ≥ 3. Jun makes f continuous. As (a, b), what are a and b?

    (3, 0)

  4. f(0) = 5; f(1) = 8; f(2) = -2; f(3) = -6; f(4) = 3. Luz studies the table shown. What does the IVT say about zeros of f between x = 0 and x = 4?

    • no zero is promised
    • exactly 2 zeros
    • at least 2 zeros

  5. f(0) = -7; f(1) = 4; f(2) = 6; f(3) = 9; f(4) = 12. Milo studies the table shown. What does the IVT say about zeros of f between x = 0 and x = 4?

    • at least 1 zero
    • exactly 1 zero
    • no zero is promised

  6. f(0) = -8; f(1) = 3; f(2) = -5; f(3) = 7; f(4) = -9. Noah studies the table shown. What does the IVT say about zeros of f between x = 0 and x = 4?

    • exactly 4 zeros
    • at least 3 zeros
    • at least 4 zeros

  7. The air is 12 °C at 6 a.m. and 25 °C at noon, changing continuously. Pia asks whether it is ever exactly 20 °C. Which is true?

    • yes, if it changes at a steady rate
    • not promised: 20 °C is outside the range
    • yes, at some moment it equals 20 °C

  8. A drone climbs continuously from 20 m to 80 m in 10 seconds. Ren asks whether it is ever exactly 95 m. Which is true?

    • yes, at some moment it equals 95 m
    • yes, if it changes fast enough
    • not promised: 95 m is outside the range

  9. Soup cools continuously from 90 °C to 40 °C in an hour. Tala asks whether it is ever exactly 60 °C. Which is true?

    • yes, if it changes at a steady rate
    • not promised: 60 °C is outside the range
    • yes, at some moment it equals 60 °C

  10. A pool's depth drains continuously from 5 ft to 2 ft in a day. Zee asks whether it is ever exactly 1 ft. Which is true?

    • yes, at some moment it equals 1 ft
    • not promised: 1 ft is outside the range
    • yes, if it changes fast enough