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Function Notations and Operations
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Even and Odd Functions

Example

Identify if each function is even, odd, or neither even nor odd.

1.  f(x)=x
2.  f(x)= x 2 +9
3.  f(x)=2 x 3 +5


Solution
1. Given   f(x)=x, find   f(−x).
f(−x)=−x

Notice that   f(x)≠f(−x). Thus,   f(x) is not an even function. However,   f(−x)=−f(x). Thus,   f(x) is an odd function.
2. Given   f(x)= x 2 +9, find   f(−x).
f(−x)= (−x) 2 +9= x 2 +9

Notice that   f(x)=f(−x). Thus,   f(x) is an even function.
3. Given   f(x)=2 x 3 +5, find   f(−x).
f(x)=2 (−x) 3 +5=−2 x 3 +5

Notice that   f(x)≠f(−x) and   f(−x)≠−f(x). Thus,   f(x) is neither an even function nor an odd function.