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

Problem

ƒ(x)=x3−x

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 for every instance of x in the original function.

ƒ*(−x)=(−x)3−(−x)

3. Simplify the expression by applying the properties of exponents and signs. Since (−x)3=−x3 and −(−x)=x we rewrite the expression.

ƒ*(−x)=−x3+x

4. Factor out a negative sign from the simplified expression to compare it to the original function.

ƒ*(−x)=−(x3−x)

5. Compare the result to the original function ƒ(x) Since ƒ*(−x)=−ƒ(x) the function satisfies the condition for an odd function.

−(x3−x)=−ƒ(x)

Final Answer

ƒ(x)=Odd

Want more problems? Check here!