Determine if Odd, Even, or Neither f(x)=x^3-5

Problem

ƒ(x)=x3−5

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 for every instance of x

ƒ*(−x)=(−x)3−5

3. Simplify the expression by applying the power to the negative sign. Since the exponent is odd, (−x)3=−x3

ƒ*(−x)=−x3−5

4. Compare the result to the original function ƒ(x)=x3−5 Since −x3−5≠x3−5 the function is not even.

5. Compare the result to the negative of the original function −ƒ(x)=−(x3−5)=−x3+5 Since −x3−5≠−x3+5 the function is not odd.

6. Conclude that because the function satisfies neither condition, it is neither even nor odd.

Final Answer

ƒ(x)=x3−5* is neither even nor odd

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