How do you determine whether $f (x) = x^{5} - x$ $f(x)=x^5-x$ is even, odd, or neither?

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Answer 1 · 2017-01-06T07:27:22

The function is odd

$f (x) = x^{5} - x$ $f(x)=x^5-x$

We calculate

$f (- x) = - x^{5} + x$ $f(-x)=-x^5+x$

$- f (x) = - (x^{5} - x) = - x^{5} + x$ $-f(x)=-(x^5-x)=-x^5+x$

Therefore,

$f (- x) = - f (x)$ $f(-x)=-f(x)$

The function is odd, that is, symmetric wrt the origin $O$ $O$

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

graph{x^5-x [-4.385, 4.385, -2.19, 2.194]}

How do you determine whether f(x)=x^5-xf(x)=x5−xf(x)=x^5-x is even, odd, or neither?