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Numerade Educator

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Problem 65 Hard Difficulty

Suppose $ g $ is an even function and let $ h = f \circ g $. Is $ h $ always an even function?

Answer

We need to examine $h(-x)$.
$h(-x)=(f \circ g)(-x)=f(g(-x))=f(g(x)) \quad \text { [because } g \text { is even }] \quad=h(x)$
Because $h(-x)=h(x), h$ is an even function.

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Video Transcript

Okay, So suppose g's even and let h equal atthe composed with G. Question is, is h always an even function? Okay, so I just wrote down, you know, a simple definition. Um, I'm trying to see it. You know, each of negative X is h of X, then we haven't even function. Or if it equals minus h of X, we have an odd function, but in this case Ah, this doesn't really help us out, because, jeez, even And so we'LL get this equality on the rainy inside. But, you know, ah, this doesn't really say anything about the overall function. But if we take a case, you know, case one where, um f is and even degree polynomial see like, see, like a power function x to the power and, you know, and where and is even And of course, G is also even is well, And let's call that accident em where m iss even a swell. So this composition here will turn into ah x on Jerry. Let's let's fill this out to get death of X Teo the Power em Jason. Because that's what Jia's and let's evaluate death of exit e m and we'LL get X to the a m to the power end and using our laws of exponents This will be X to the M times and and noticed that and is even an M iss Even so, the product of m times in this even Okay, thus we can say that Ah, In this case, if f isn't even agreed that this composition therefore is even Okay, So this implies that s Circle G is even. But now we need to consider another case where f is an odd degree polynomial this time or some other Ah, our function. So let's look at case too. He's to say f is odd degree of polynomial where the f is next to the power we said and and and is an odd number in GI is still excellent m and as given and is even Okay, So this composition f of g of ax it's going to be f of X to the Power M where Emma's even and finally this composition will be X to the M to the power end and using our lots of exponents. This's again X times, um, extra m times and okay, And take note that No, that m times. And okay, where m is an even number and is in our DNA number. Hey, so this product is even odd. Times even isn't even number. Hey, so because of that, this entire being me this entire function, it's even so f circle chief ISS even And either case, where s is even or F is odd. He therefore half circle g iss even whether ethics, even or odd.