How do you determine if f(x)=8x^6 + 12x^2 is an even or odd function?

2 Answers
Mar 20, 2016

It's an even function.

Explanation:

One of the easy ways to determine weather a function is even or odd is to look at the powers of x. If all the powers of x are even, such as 8x^6+12x^2 (powers are 6 and 2), then it's an even function. If all the powers of x are odd, such as 5x^3+x (powers of x are 3 and 1) , then it's an odd function.

Also remember that a function can be neither odd or even function, such as;
f(x)=5x^3+x^4

Mar 20, 2016

Verify f(-x) = f(x) for all x in RR, so f(x) is even.

Explanation:

f(-x) = 8(-x)^6+12(-x)^2 = 8x^6+12x^2 = f(x) for all x in RR

So f(x) is even.

For polynomial functions, there is a quick shortcut:

If all of the terms have even degree then the function is even. Remember that a constant term is of degree 0 which is even.

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

If the terms are a mixture of odd and even degrees then the function is neither even nor odd.