Question

Is the trig function: even, odd, neither `y=x-sin(x)`?

Answer 1

Odd function

Explanation:

For even functions, `f(x)=f(-x)`
For odd functions, `f(-x)=-f(x)`

Let `f(x)=x-sinx`

To prove it is an even function
`f(-x)=(-x)-sin(-x)`
`f(-x)=-x+sinx`
Therefore, `f(x)!=f(-x)` so it is not an even function

Following on from the last step,
`f(-x)=-(x-sinx)`
`f(-x)=-f(x)`
Therefore, it is an odd function

Answer 2

The function is odd.

Explanation:

Let `f(x)=x-sinx`

A function is even if `f(-x)=f(x)`

A function is even if `f(-x)=-f(x)`

Therefore,

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

As `sin(-x)=-sinx`

`f(-x)=-x+sinx=-(x-sinx)=-f(x)`

The function is odd. It is symmetric about the origin.

graph{x-sinx [-16.02, 16.01, -8.01, 8.01]}