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Even and Odd FunctionsStudents will determine if a function is even,

odd, or neither using algebraic methods.

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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.

Even and Odd Functions

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• A function f is even if the graph of f is symmetric with respect to the y-axis.

Even Functions

f(x) = |x| - 3 f(x) = |x + 6|

Even Not an Even

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• 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)

Using Algebraic Method

f(x) = |x + 6| f(x) = |x| - 3

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• A function f is odd if the graph of f is symmetric with respect to the origin.

Odd Functions

Odd Function Not Odd Function

f(x) = 3x f(x) = 3x + 6

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• 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

Using Algebraic Method

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• 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.

Example

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• 1)  f(x) = x³ + x² + x + 1Even 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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