determining whether a function is even or odd

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how do you determine whether a function is even or odd (or neither)?

for example: f(x) = -x^6 + 2x^2 + 2

thanks!
 
For polynomials, you just have to look at it. Are all the exponents even numbers? Even function.

In your example, you have 6, 2, and 0. Those are all even.
 
For a technically-correct solution (if your course requires that you "show your work"), just follow the formula they gave you in your book: Plug "-x" in for every "x", and simplify. If you end up with f(x) (that is, with exactly what you started with), then the function is even; if you end up with -f(x) (that is, with the same thing, but with all the signs changed), then the function is odd; otherwise, the function is neither.

Eliz.
 
thanks

thanks, i tried that formula and i got and negative x for -x^6 (b/c -(-x)^6) and i got a positive 2x. does this make the equation neither then since they're not all even or odd?

sorry, i feel so stupid b/c we haven't even gotten our books yet (this is just summer homework and we have nothing to teahc us or even help us except the internet). but thanks!
 
I'm sorry, but I do not understand your reply. How did "x^6" turn into "2x"? What do you mean by "all even or odd"? Do you mean "positive or negative"? Please clarify.

Eliz.
 
oh wow i'm sorry i meant all positive or negative. i feel dumb lol.

when i plug in -x to f(x) = -x^6 + 2x^2 + 2 i get:

f(x) = -(-x)^6 + 2(-x)^2 + 2

this makes the first one negative, the second positive, and the 2 of course stays positive. so since there's not all either negative or positives, would this mean that the problem is considered neither instead of even or odd?

really sorry about the last post. thanks!
 
Please review my earlier reply. I did not say that f(-x) would have "all negative terms" or "all positive terms", but that all the terms would have opposite signs from f(x) (if the function were odd) or all the exact same signs as f(x) (if the function were even).

Write down f(x).

Evaluate f(-x), and simplify fully.

Compare the results. Specifically, look at the signs on each term. How do f(x) and f(-x) compare?

Eliz.
 
ok

ok so if only one of terms out of the three has an opposite sign does that mean that it's considered neither? i'm just not sure on when it's considered neither or not.

thanks!
 
If ALL the signs have changed, the function is odd. If NONE of the signs has changed, the function is even. In any other case, the function is neither.

Eliz.
 
got it

ok that's what i thought. thank you so much! i'm usually a lot smarter than this, my brain is just fried from hours of pre calc homework and not being in school for 3 months to remember anything. thanks again!
 
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