How do you determine if # h(x)= x^7+x^3+7# is an even or odd function?

1 Answer
George C.
May 24, 2016

It is neither.

Explanation:

The shortcut method is to note that #h(x)# is a polynomial with terms with a mixture of odd degree (#x^7# and #x^3#) and even degree (#7#), so it is neither an odd nor even function.

  • An even function satisfies #f(-x) = f(x)# for all #x# in the domain.

  • An odd function satisfies #f(-x) = -f(x)# for all #x# in the domain.

We find:

#h(1) = 1+1+7=9#

#h(-1) = -1-1+7 = 5#

So #h(x)# satisfies neither condition.

In general, one effective method to tell whether a function #f(x)# is odd, even or neither is to substitute #-x# for #x# into its definition and see if the result simplifies to #f(x)# or #-f(x)#.

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