Odd function by definition has the property:
[tex] f(-x)=-f(x) [/tex]
Thus, we can immediately rule out all even powers of x. Now, we are left with the second and the last option. We can rule out the second option too because [tex] f(x)=4x^3+7 [/tex] and thus,
[tex] f(-x)=-4x^3+7 \neq-f(x) [/tex]
Now, the last option is: [tex] f(x) = 6x^3 + 2x [/tex]
[tex] \therefore f(-x) = 6(-x)^3 + 2(-x)=-6x^3-2x=-(6x^3 + 2x)=-f(x) [/tex]
[tex] \because f(-x)=-f(x) [/tex], therefore, the function in the last option is an odd function.
