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Problem 50 Hard Difficulty

Express the function in the form $ f \circ g \circ h $.

$ H(x) = \sqrt[8]{2 + \mid x \mid} $

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Video Transcript

here we see the function capital H of X, and we want to break that down into a composition f of G of h. Another way to write that is with parentheses. F of G of h of x. Okay, so that shows us that H of X is going to be the innermost function. So if we look back at capital age, it looks like absolute value of X could be the innermost function. So let's let little h of X equal the absolute value of X. Now the question is, what is it inside of? Because that has to be inside of G. So let's figure out g of X. If we let g of X equal to plus X, then it looks like a chav X was put in there in the place of X, so that G of h of X would be to plus the absolute value of X. Now. Thirdly, we want to find the outermost function F. So what is surrounding G of h of X? It's being surrounded by the eighth route. So if f of X is the eighth route of X and if we substitute G of h of X into it, then we're substituting two plus absolute value backs into the eighth route and we do get capital age. That's what we were hoping for.