Determine if Odd, Even, or Neither f(x)=x^3

Problem

ƒ(x)=x3

Solution

1. Recall the definitions for function parity: a function is even if ƒ*(−x)=ƒ(x) and odd if ƒ*(−x)=−ƒ(x)

2. Substitute −x into the function in place of x

ƒ*(−x)=(−x)3

3. Simplify the expression by applying the power to the negative sign.

ƒ*(−x)=(−1)3⋅x3

ƒ*(−x)=−x3

4. Compare the result to the original function ƒ(x)=x3

−x3=−ƒ(x)

5. Conclude that since ƒ*(−x)=−ƒ(x) the function is odd.

Final Answer

ƒ(x)=x3* is Odd

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