Question

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

Answer 1

The function will be odd.

Explanation:

For an even function, `f(-x) = f(x)`.
For an odd function, `f(-x) = -f(x)`

So we can test this by plugging in `x = -x`:
`-x - sin(x) = -x + sin(x) = (-1)(x - sin(x))`

This means the function must be odd.

It's not surprising either, since `x` and `sin(x)` are both odd. In fact, given two functions, `f(x)` and `g(x)` for which:
`f(-x) = -f(x)`
`g(-x) = -g(x)`

It is obvious that:
`f(-x) + g(-x) = -f(x) - g(x) = -[f(x) + g(x)]`

That is, the sum of odd functions is always another odd function.

Answer 2

`f(x)=x-sinx` is odd

Explanation:

A function `f` is said to be even if `f(-x)=f(x)`, and odd if `f(-x)=-f(x)`. Then, to check, we will evaluate the function applied to `-x`.

In our case, `f(x)=x-sinx`, so

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

`=-x-(-sinx)` (as `sinx` is odd)

`=-x+sinx`

`=-(x-sinx)`

#=-f(x)

Thus `f(x)=x-sinx` is odd.