How do you determine if #f(x) = x^2 - x^8# is an even or odd function?

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Answer 1 · 2016-04-19T03:34:22

This is an even function.

If a function is even, #\quad f(-x)=f(x)#

If a function is odd, #\quad f(-x)=-f(x)#

Visibly, this means that if we flip an even function about the #y# axis, we get the same function back. It looks the same on both sides.

If we were to reflect an odd function about the #y# axis, it would look the same but flipped upside down.

How to determine if a function is even or odd?

#Try\quad # substituting #(-x)# in the place of #(x)# and see what we get.

#f(-x)=(-x)^2-(-x)^8#

but real numbers raised to even powers are always positive, so the sign doesn't matter ex) #(-x)^2=x^2#

Therefore,
#f(-x)=(-x)^2-(-x)^8=x^2-x^8=f(x)#

It turns out to be the exact same as the original function. Because

#f(-x)=f(x)#, this is an even function.