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Problem 49 Medium Difficulty

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

$ R(x) = \sqrt{ \sqrt{x} - 1} $

Answer

Let $h(x)=\sqrt{x}, g(x)=x-1,$ and $f(x)=\sqrt{x} .$ Then
\[
(f \circ g \circ h)(x)=f(g(h(x)))=f(g(\sqrt{x}))=f(\sqrt{x}-1)=\sqrt{\sqrt{x}-1}=R(x)
\].

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Video Transcript

here we have the function R of X, and we're going to think of it as a composition f of G of h. Another way to write that would be f of G of h of X. So basically, we want to figure out what f g and h r To make this work. H will be the innermost function. And the innermost function we see is the square root of X. So let's let h of x equals the square root of X. Now we need the middle A or G G would be what h of x was plugged into. So if we let g of x b X minus one, we could see that if we were to calculate G of h of X, we would get square root of X minus one. So that's inside the outside radical. So that means the outermost function f is the outside radical. We'll call that square Rydex. So you see, if we were to put G of h of X inside a bath, we would be putting the square root of X minus one inside a square root. And that was our goal. That was what we were trying to get