Question

Which of the following is an odd function?

f(x) = 3x2 + x
f(x) = 4x3 + 7
f(x) = 5x2 + 9
f(x) = 6x3 + 2x

Answer 1

Odd function by definition has the property:

`f(-x)=-f(x)`

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
`f(x)=4x^3+7` and thus,

`f(-x)=-4x^3+7 \\eq-f(x)`

Now, the last option is:
`f(x) = 6x^3 + 2x`

`\therefore f(-x) = 6(-x)^3 + 2(-x)=-6x^3-2x=-(6x^3 + 2x)=-f(x)`

`\because f(-x)=-f(x)`, therefore, the function in the last option is an odd function.