Question

How do you determine if `x-sin(x)` is an even or odd function?

Answer 1

Examine how it acts on `-x` to find that `f(x)=x-sin(x)` is an odd function

Explanation:

A function `f` is called even if `f(-x) = f(x)` and odd if `f(-x)=-f(x)`. Then, to determine whether the given function is even, odd, both, or neither, we examine how it acts on `-x`.

Letting `f(x) = x-sin(x)`, we have

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

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

`=-x+sin(x)`

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

`=-f(x)`

Thus, `f(x)=x-sin(x)` is an odd function#