Determine if Odd, Even, or Neither g(x)=-4x^5+7x^2

Problem

g(x)=−4*x^5+7*x^2

Solution

1. Recall the definitions for function symmetry: a function is even if g*(−x)=g(x) odd if g*(−x)=−g(x) and neither if neither condition is met.

2. Substitute −x for every x in the original function to evaluate g*(−x)

g*(−x)=−4*(−x)^5+7*(−x)^2

3. Simplify the powers of the negative variable, noting that (−x)^5=−x^5 and (−x)^2=x^2

g*(−x)=−4*(−x^5)+7*(x^2)

g*(−x)=4*x^5+7*x^2

4. Compare the result to the original function g(x)=−4*x^5+7*x^2 Since 4*x^5+7*x^2≠−4*x^5+7*x^2 the function is not even.

5. Compare the result to the negative of the original function −g(x)=−(−4*x^5+7*x^2)=4*x^5−7*x^2 Since 4*x^5+7*x^2≠4*x^5−7*x^2 the function is not odd.

Final Answer

g(x)=−4*x^5+7*x^2* is Neither

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