Question

Which of the following is an odd function?

Answer 1

Answer:

F(x)=6x^3+2x

Step-by-step explanation:

Odd functions have solely odd powers of x.  

F(x)=4x^3+7      is actually   F(x) = 4x^3 + 7x^0, which is neither even nor odd.

F(x)=6x^3+2x   has only odd powers of x:  x^3 and x^1.  This is the answer.

Answer 2

Answer:

$\large\boxed{f(x)=6x^3+2x}$

Step-by-step explanation:

$\text{If}\ f(-x)=f(x)\ \text{then}\ f(x)\ \text{is an even function.}\\\\\text{If}\ f(-x)=-f(x)\ \text{then}\ f(x)\ \text{is an odd function.}$

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$f(x)=3x^2+x\\\\f(-x)=3(-x)^2+(-x)=3x^2-x\\\\f(-x)\neq f(x)\ \wedge\ f(-x)\neq-f(x)\\\\============================\\\\f(x)=4x^3+7\\\\f(-x)=4(-x)^3+7=-4x^3+7\\\\f(-x)\neq f(x)\ \wedge\ f(-x)\neq-f(x)\\\\============================\\\\f(x)=5x^2+9\\\\f(-x)=5(-x)^2+9=5x^2+9\\\\f(-x)=f(x)-\text{It's an even function}\\\\============================\\\\f(x)=6x^3+2x\\\\f(-x)=6(-x)^3+2(-x)=-6x^3-2x=-(6x^3+2x)\\\\f(-x)=-f(x)-\text{It's an odd function.}$