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

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Answer 1 · 2018-06-16T11:53:11

Odd function

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 · 2018-06-16T11:58:03

The function is odd.

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]}