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

2 Answers
Jul 20, 2016

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.

Jul 20, 2016

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.