Piecewise continuity and the Intermediate Value Theorem

Answer Key - Grade 12 Math Worksheet - Building Confidence

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

1. 1.

   f(x) = 2x + 4 for x < 1 and f(x) = 5x + 3 for x ≥ 1. Bo checks the join. What happens at x = 1?

   • a hole at x = 1
   • a jump at x = 1
   • continuous at x = 1
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. 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
   • a hole at x = -1
   • continuous at x = -1
4. 4.

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

   9
5. 5.

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

   4
6. 6.

   f(x) = x + 5 for x < 4 and f(x) = 3x − 2 for x ≥ 4. Milo needs f(4). What is it?

   10
7. 7.

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

   7
8. 8.

   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]
9. 9.

   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]
   • yes: f must cross 0 between 1 and 6
   • no zero is possible on [1, 6]
10. 10.

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

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