How do you determine if #4x^5 / absx # is an even or odd function?

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Answer 1 · 2018-06-07T19:24:39

The function is odd.

We have

#f(x)=(4x^5)/|x|#

We wish to find if #f# is even or odd. To make sure, let's remeber what those things mean:

#{(f " is even" => f(-x)=f(x)),( f " is odd" => f(-x)=-f(x)) :}#

Note that in both definitions we had #f(-x)# being compared to #f(x)#. Let's do the same.

#f(x)=(4x^5)/|x|#
#f(-x)=(4(-x)^5)/|-x|#

For real numbers, #|a|=|-a|# by the definition of the absolute value.

Now, because #x^5# has an odd exponent, #(-x)^5=-x^5#. For a general rule, if #k# is odd then

#(-x)^k=-x^k#

Hence,

#f(-x)=-(4x^5)/|x|=-f(x)#

We see now that #f# is an odd function.