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

Problem

g(x)=−4*x5+7*x2

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=−x5 and (−x)2=x2

g*(−x)=−4*(−x5)+7*(x2)

g*(−x)=4*x5+7*x2

4. Compare the result to the original function g(x)=−4*x5+7*x2 Since 4*x5+7*x2≠−4*x5+7*x2 the function is not even.

5. Compare the result to the negative of the original function −g(x)=−(−4*x5+7*x2)=4*x5−7*x2 Since 4*x5+7*x2≠4*x5−7*x2 the function is not odd.

Final Answer

g(x)=−4*x5+7*x2* is Neither

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