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Problem 79 Hard Difficulty

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

Answer

(i) If $f$ and $g$ are both even functions, then $f(-x)=f(x)$ and $g(-x)=g(x) .$ Now $(f+g)(-x)=f(-x)+g(-x)=f(x)+g(x)=(f+g)(x),$ so $f+g$ is an even function.
(ii) If $f$ and $g$ are both odd functions, then $f(-x)=-f(x)$ and $g(-x)=-g(x) .$ Now $(f+g)(-x)=f(-x)+g(-x)=-f(x)+[-g(x)]=-[f(x)+g(x)]=-(f+g)(x),$ so $f+g$ is an odd function
(iii) If $f$ is an even function and $g$ is an odd function, then $(f+g)(-x)=f(-x)+g(-x)=f(x)+[-g(x)]=f(x)-g(x)$
which is not $(f+g)(x)$ nor $-(f+g)(x),$ so $f+g$ is neither even nor odd. (Exception: if $f$ is the zero function, then
$f+g$ will be odd. If $g$ is the zero function, then $f+g$ will be even. .

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Video Transcript

If F isn't even function, then we know that opposite X values have the same Y value, so f of X equals f of the opposite of X. If G is even then, opposite X values have the same y value. So we know that JIA Becks equals G of the opposite of X. So then what would have plus g b half of X plus g of X? Well, we could replace f of X with f of the opposite of X since they're equal. And then we could replace G of X with G of the opposite of X since they're equal. And so now we know that f of x plus g of X is equal to f of the opposite of X plus g of the opposite of X, which means that F plus G is also even. What if f Izod and G is odd? What about F plus G? So if F is odd than F of the opposite of X will equal the opposite of F of X, we could also say that the opposite of F of the opposite of X equals f of X. That's a mouthful. If she is odd same thing. G of the opposite of X is equal to the opposite of G of X. We multiply both sides by negative one. The opposite of G of the opposite of X is equal to G of X. So what about F Plus G? Half of eggs, plus G of X could be replaced with. We could replace F of X with the opposite of F of the opposite of X. And then we could replace G of X with the opposite of G of the opposite of X. And this If we want Teoh factor out, the negative would get the opposite of f of the opposite of X plus Jeep the opposite of X. So comparing this line and this line, we see that f plus G is odd. All right, so what if one of them is even in one of them? Is that if f is even then, f of X equals the equals f of the opposite of X? If G Assad, then g of X equals the opposite of G of the opposite of X. So what about F plus G half of X plus g of X? We could replace r F of X with f of the opposite of X. We could replace R G of X, with the opposite of G of the opposite of X. So we have f of the opposite of X minus G of the opposite of X that does not look equivalent to the original. That does not look like the opposite of the original, so it's neither on nor even.