1-06 Even and Odd Functions Notes

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1-06 Even and Odd Functions Notes

  1. 1. Even and Odd FunctionsStudents will determine if a function is even, odd, or neither using algebraic methods. 1
  2. 2. Even and Odd Functions We can define a function according to itssymmetry to the y – axis or to the origin.This symmetry will also correspond withcertain Algebraic conditions. The function can be classified as eithereven, odd or neither. 2
  3. 3. Even Functions• A function f is even if the graph of f issymmetric with respect to the y-axis. Even Not an Even f(x) = |x| - 3 f(x) = |x + 6| 3
  4. 4. Using Algebraic Method • Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f. • Test Algebraically for f(2) and f(-2) f(x) = |x| - 3 f(x) = |x + 6| 4
  5. 5. Odd Functions • A function f is odd if the graph of f is symmetric with respect to the origin. Odd Function Not Odd Function f(x) = 3x f(x) = 3x + 6 5
  6. 6. Using Algebraic Method• Algebraically, f is odd if and only if f(‐x) = ‐f(x) for all x in the domain of f.• Test Algebraically for  f(‐2) and  ‐ f(2) f(x) = 3x f(x) = 3x + 6 6
  7. 7. Example• Ex. 1 Test this function for symmetry:• f(x) = x5 + x³ + x• Solution.   We must look at f(−x):• f(−x) = (−x)5 + (−x)³ + (−x)   = −x5 − x³ − x = −(x5 + x³ + x)    = −f(x)• Since  f(−x) = −f(x), this function is symmetrical withrespect to the origin.• Remember: A function that is symmetrical withrespect to the origin  is called an odd function. 7
  8. 8. Your Turn• 1)  f(x) = x³ + x² + x + 1 Even Odd• Answer:   Neither, because f(−x) ≠ f(x) , and f(−x) ≠ −f(x).• 2)  f(x) = 2x³ − 4x• Answer:   f(x) is odd. It is symmetrical withrespect to the origin because f(−x) = −f(x).• 3)  f(x) = 7x² − 11• Answer:   f(x) is even -- it is symmetrical withrespect to the y-axis -- because f(−x) = f(x). 8

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