Question 400142
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ \frac{x\ -\ 1}{x}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ g(x)\ =\ \sqrt{x\ +\ 3}]


We want to determine


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f\left(f\left\[g\left(x\right)\right\]\right)]


First determine


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f\left(g\left(x\right)\right)\ =\ \frac{g(x)\ -\ 1}{g(x)}\ =\ \frac{\sqrt{x\,+\,3}\ -\ 1}{\sqrt{x\,+\,3} ]


Then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f\left(f\left\[g\left(x\right)\right\]\right)\ =\ \frac{\frac{\sqrt{x\,+\,3}\ -\ 1}{\sqrt{x\,+\,3}}\ -\ 1}{\frac{\sqrt{x\,+\,3}\ -\ 1}{\sqrt{x\,+\,3}}]


Simplify:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f\left(f\left\[g\left(x\right)\right\]\right)\ =\ \frac{1}{1\ -\ \sqrt{x\,+\,3}]


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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