How do you determine if #f(x) = 3x^4 - x^2 + 2# is an even or odd function?

1 Answer
George C.
Jul 28, 2016

This #f(x)# is an even function.

Explanation:

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

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

In the case of #f(x) = 3x^4-x^2+2# we find:

#f(-x) = 3(-x)^4-(-x)^2+2 = 3x^4-x^2+2 = f(x)#

So #f(x)# is an even function.

Actually, there is a shortcut with polynomial functions:

  • If all of the terms have even degree then the function is even.

  • If all of the terms have odd degree then the function is odd.

  • Otherwise the function is neither odd nor even.

In our example, #color(blue)(3x^4)# has degree #4#, #color(blue)(-x^2)# has degree #2# and #color(blue)(2)# has degree #0#. So all of the terms are of even degree and the function is even.

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