Evaluating Piecewise Functions

Introduction

• Topic of focus: Evaluating piecewise functions
• Request from many students for additional content on this topic.

Definition

• Piecewise function: A function defined in pieces or segments.
• Example structure:
  • f(x) = -x - 4, if x ≤ 5
  • f(x) = 2x² - x, if 5 < x ≤ 10

Example 1: Evaluating f(-2) and f(7)

Step 1: Evaluate f(-2)

• Check the condition for -2:
  • Since -2 < 5, use f(x) = -x - 4.
• Substitute -2 into the function:
  • f(-2) = -(-2) - 4
  • = 2 - 4
  • = -2
• Conclusion: f(-2) = -2

Step 2: Evaluate f(7)

• Check the condition for 7:
  • Since 5 < 7 ≤ 10, use f(x) = 2x² - x.
• Substitute 7 into the function:
  • f(7) = 2(7)² - 7
  • = 2(49) - 7
  • = 98 - 7
  • = 91
• Conclusion: f(7) = 91

Example 2: Evaluating f(-5) and f(1)

Step 1: Evaluate f(-5)

• Check the condition for -5:
  • Since -5 ≤ 0, use f(x) = 14.
• Conclusion: f(-5) = 14

Step 2: Evaluate f(-3)

• f(-3) is also ≤ 0, hence f(-3) = 14
• Conclusion: f(-3) = 14

Step 3: Evaluate f(1)

• Check the condition for 1:
  • Since 0 < 1 < 15, use f(x) = x² - 9x.
• Substitute 1 into the function:
  • f(1) = 1² - 9(1)
  • = 1 - 9
  • = -8
• Conclusion: f(1) = -8

Conclusion

• Understanding piecewise functions involves analyzing conditions and substituting values appropriately.
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