Problem 66 Hard Difficulty

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

Answer

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Video Transcript

suppose G is an odd function and let h equal f compose with G. And the question is, is each always in our function? What if f is on? And what if f is even? Okay, So I think this question relies entirely on the last two questions. Whether at this odd or at this even so, let's look at a case when f is odd, for example. Okay. And then we'LL let maybe s equal X to the power and where in is hot and odd number So we have a non degree power function Here s o we end up with Oh, and let's ah, let g equal x the power m where and is odd because G is not function as given. Okay, so f complies with G There's going to be s of exit e m where in is odd And now this compass is composition will be excellent m to keep our end in using laws exponents, we have extra m times end and notice that the product m times end if an Izod in Amazon than and the product m times and is on his well, so thus we have S H is our But now we got a look at the second case where f is an even function. Okay. And, um will assume the above except, uh, for N is even function or is even number Then we get half composed with G is X sorry f to the X to power em. This is equal to let me write about eleven more clear And this is X to the M to the power n and lost exponents saying this is exito m times and and know that and plans And if m is he said, And this odd and n is even me and of getting even No. Okay, so why and you know, live in one room so f compose with G is even yeah versus the first case, which we said after both ji Assad. So therefore, to answer the question is each always non function. Therefore, age is not always even or you could say we're on. It all depends on the degree of s or media. If f is even Iran