How do you determine if f(x)=8x^6 + 12x^2f(x)=8x6+12x2 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 xx. If all the powers of xx are even, such as 8x^6+12x^28x6+12x2 (powers are 66 and 22), then it's an even function. If all the powers of xx are odd, such as 5x^3+5x3+xx (powers of xx are 33 and 11) , 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^4f(x)=5x3+x4

Mar 20, 2016

Verify f(-x) = f(x)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.