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

Jun 16, 2018

Odd function

#### Explanation:

For even functions, $f \left(x\right) = f \left(- x\right)$
For odd functions, $f \left(- x\right) = - f \left(x\right)$

Let $f \left(x\right) = x - \sin x$

To prove it is an even function
$f \left(- x\right) = \left(- x\right) - \sin \left(- x\right)$
$f \left(- x\right) = - x + \sin x$
Therefore, $f \left(x\right) \ne f \left(- x\right)$ so it is not an even function

Following on from the last step,
$f \left(- x\right) = - \left(x - \sin x\right)$
$f \left(- x\right) = - f \left(x\right)$
Therefore, it is an odd function

Jun 16, 2018

The function is odd.

#### Explanation:

Let $f \left(x\right) = x - \sin x$

A function is even if $f \left(- x\right) = f \left(x\right)$

A function is even if $f \left(- x\right) = - f \left(x\right)$

Therefore,

$f \left(- x\right) = - x - \sin \left(- x\right)$

As $\sin \left(- x\right) = - \sin x$

$f \left(- x\right) = - x + \sin x = - \left(x - \sin x\right) = - f \left(x\right)$

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

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