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Problem 36 Hard Difficulty

Find the function (a) $ f \circ g $, (b) $ g \circ f $, (c) $ f \circ f $, and (d) $ g \circ g $ and their domains.

$ f(x) = \sin x $ , $ g(x) = x^2 + 1 $

Answer

a) $f(g(x))=\sin \left(x^{2}+1\right)$
$D :(-\infty, \infty)$
b) $g(f(x))=(\sin x)^{2}+1$
$D :(-\infty, \infty)$
c) $f(f(x))=\sin (\sin x)$
$D :(-\infty, \infty)$
d) $g(g(x))=x^{4}+2 x^{2}+2$
$D :(-\infty, \infty)$

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Video Transcript

Let's start by finding f G. So we're going to take the G function and substituted into the F function in place of X, and we get the sign of the quantity X squared plus one. There's nothing we could do to simplify that. Now it's moved to part F or to Part B, so we're going to put the F function inside the G function. So that's going to look like a sign of X. Thank you going in for X in the G function, so we'll have the sign of X quantity squared plus one. Now the way we generally right the sign of X quantity squared is we write sine squared x So we have signs Word X plus one. Now let's think about the domains. So the domain of F was all real numbers. The domain of a sine function or a coastline function is all real numbers, and the domain of G is all real numbers. It's a polynomial, and the domain of any polynomial is all real numbers. So if we have a domain, if we have a function with domain, all real numbers inside of function with domain, all real numbers, there are no restrictions. So the domain of F of G is all real numbers. And the domain of GF is also all real numbers. And if you prefer, you can write that as negative Infinity to infinity. Okay, now it's do f of F and G of G. So for FF, we're going to put sign of X inside sign of X. So sign of X goes in place of X. So that gives us the sign of the sign of X m for G of G. We're going to put X squared plus one inside X squared, plus one in place of X. So we get X squared plus one quantity squared plus one that we can simplify. We can go ahead and use a foil technique to multiply X squared plus one by itself. And that gives us X to the fourth power plus two x squared plus one. And then we have the extra plus one on the end. So that's extra. The fourth power plus two x squared plus two. Now, considering the domains, we have the same sort of situation where we have all real numbers as the domain of each individual function. So there's no restriction on the composed function. All real numbers Negative. Infinity to infinity. In both cases and particularly here, we just have a polynomial for part D. So every polynomial is domain is all real numbers.

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