Is y=2x^3 + 4x an odd function, even, both, or neither?

Question

Is y=2x^3 + 4x an odd function, even, both, or neither?

Answer 1

f(x) = 2x^3 + 4x

=> f(x)

[Edit: Sorry, I meant f(-x). Thanks to Math_kp who pointed out my error.]

= 4(-x)^3 + 4(-x)

= - 4x^3 - 4x

= - f(x)

=> f(-x) = - f(x)

=> given function is an odd function.

Answer 2

odd means it is symmetric to the origin. The test for odd is...

if f(x) = - f(-x) it's odd. so you have...

f(x) = 2x³ + 4x

-f(-x) = - (2(-x)³ + 4(-x)) = -(-2x³ - 4x) = 2x³ + 4x ← same as f(x)...

it's odd!

the even test is...

if f(x) = f(-x) it's even

even means symmetric to the y-axis. If both tests fail, it is neither

Answer 3

f(x) = 2x^3 + 4x



f(-x) = 2(-x)^3 + 4(-x)

f(-x) = - 2x^3 - 4x

f(-x) = - f(x)

So it's an odd function because f (-x) = -f(x), e.g. sin(-x) = -sin(x)

If it were an even function then f(-x) = f(x), e.g. cos(-x) = cos(x)