Determine if Odd, Even, or Neither f(x)=x^3+0.04x^2+3

Problem

ƒ(x)=x3+0.04*x2+3

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+0.04*(−x)2+3

3. Simplify the expression by applying the powers to the negative signs.

ƒ*(−x)=−x3+0.04*x2+3

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

−x3+0.04*x2+3≠x3+0.04*x2+3

⇒ƒ*(−x)≠ƒ(x)

5. Compare the result to the negative of the original function −ƒ(x)=−x3−0.04*x2−3

−x3+0.04*x2+3≠−x3−0.04*x2−3

⇒ƒ*(−x)≠−ƒ(x)

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

Final Answer

ƒ(x)=Neither

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