Determine if the function f(x)=xsin^3(x) is even odd or neither. Show full work.

Question

f(x)=xsin³(x)

I think it is odd. I am not sure. I know how to do even and odd functions but not with sin. I am confused. Please help. Thank you.

Answer 1

Graphing this function we see it looks like this:

A function that is symmetrical across the y-axis (like y = x²) is even. A function that is symmetrical around the origin (like y = x³) is odd. We can see this function is symmetrical across the y-axis so the answer will be that it is even. But this is not the explanation that is being asked for. To get that, we use the definition of even and odd functions.

For Even Functions, f(x) = f(-x) This means that if I go, from the origin, a certain number of units to the right or that same number of units to the left, the y-value, aka f(x), will be the same (or the function is symmetrical across the y-axis)

For Odd Functions, f(-x) = -f(x) This means that if I go, from the origin, a certain number of units to the left I will be at the opposite y value as if I go that same number of units the the right of the origin (or the function is flipped both in the x and y direction around the origin)

In this case:

f(-x) = (-x)sin³(-x)

f(-x) = (-x)[sin(-x)]³

Using the negative angle identity, sin(-x) = -sin(x) we get:

f(-x) = (-x)[-sin(x)]³

f(-x) = (x)[sin(x)]³

f(-x) = (x)sin³(x)

or f(x) = f(-x) so the function is Even