1. Even and Odd Functions
Students will determine if a function is even,
odd, or neither using algebraic methods.
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2. Even and Odd Functions
We can define a function according to its
symmetry to the y – axis or to the origin.
This symmetry will also correspond with
certain Algebraic conditions.
The function can be classified as either
even, odd or neither.
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3. Even Functions
• A function f is even if the graph of f is
symmetric with respect to the y-axis.
Even Not an Even
f(x) = |x| - 3 f(x) = |x + 6|
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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|
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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
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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
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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 with
respect to the origin.
• Remember: A function that is symmetrical with
respect to the origin is called an odd function.
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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 with
respect to the origin because f(−x) = −f(x).
• 3) f(x) = 7x² − 11
• Answer: f(x) is even -- it is symmetrical with
respect to the y-axis -- because f(−x) = f(x).
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