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

1 Answer
Apr 19, 2016

This is an even function.

Explanation:

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.