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Problem 80 Hard Difficulty

If $ f $ and $ g $ are both even functions, is the product $ fg $ even? If $ f $ and $ g $ are both odd functions, is $ fg $ odd? What if $ f $ is even and $ g $ is odd? Justify your answers.

Answer

$f g$ is even if both $f$ and $g$ are even, or both $f$ and $g$ are odd.
If one of them is odd and the other is even, then $f g$ is odd.

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Video Transcript

If F is even then, F of X is equal to f of the opposite of X. Opposite X values have the same way value. If G is even then, G of X is equal to G of the opposite of X. Again opposite X values have the same y value. So what about half times G f of x times G of X? Well, we could replace the F of X with f of the opposite of X, and we could replace the G of X with G of the opposite of X. So f of x g of X is equal to f of the opposite of X g of the opposite of X. That means half times G is even now. Suppose that F Izod and G. Assad, then f of X equals the opposite of F of the opposite X. Opposite X values have opposite y values when it's odd and G of X is equal to the opposite of G of the opposite of X. Again opposite X values have opposite why they'll use when it's odd. So what about F times G f of x times G of X? We could replace the F of X with the opposite of F of the opposite of X. And we could replace G of X with the opposite of G of the opposite of X notice that we have two negatives. Multiply those together and we get a positive. So now we have f of the opposite of X times g of the opposite of X. That looks like the same as and even function that we just saw in party. And now suppose that f is even Angie Izzat, if ever is even than F of X is equal to f of the opposite of X Opposite X values have the same way value. If G. Assad, then JIA Becks is equal to the opposite of G of the opposite of X opposite X values have opposite y values. So what about F Time Street? F of X Times G of X would be equal to f of the opposite of X times the opposite of G of the opposite of X. Let's bring that negative sign out to the front and this will be equal to the opposite of f of the opposite of X times G of the opposite sex. Okay, so these are opposites. Therefore, f times G is odd