for the given functions f,g,and h, find fogoh and state the exact domain of fogoh, show all work  f (x)= sqtx   g(x)= e x-13e x/2+42 h (x)=2xneed answer asap

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that (fog)(x) = f(g(x)) such that:

`f(g(x)) = f(e^x - 13sqrt(e^x) + 42)`

Hence, you notice that you need to substitute x in equation of f(x) by equation `e^x - 13sqrt(e^x) + 42`  such that:

`f(e^x - 13sqrt(e^x) + 42) = sqrt(e^x - 13sqrt(e^x) + 42)`

`(fogoh) = f(g(h(x)))`

Hence, you need to substitute x in `sqrt(e^x - 13sqrt(e^x) + 42)`  by h(x) such that:

`f(g(h(x))) = sqrt(e^(h(x)) - 13sqrt(e^(h(x))) + 42)`

`f(g(h(x))) = sqrt(e^(2x) - 13sqrt(e^(2x)) + 42)`

`f(g(h(x))) = sqrt(e^(2x) - 13e^(x) + 42)`

The domain of function f(g(h(x))) comprises all x values such that `e^(2x) - 13e^(x) + 42 gt= 0` .

You need to solve the equation `e^(2x) - 13e^(x) + 42`  = 0.

You should come up with the substitution `y=e^x`  such that:

`y^2 - 13y + 42 = 0`

`y_(1,2) = (13 +- sqrt(169-168))/2 =gt y_(1,2) = (13 +-1)/2`

`y_1 = 7, y_2 = 6` Hence `e^x = 7 =gt x = ln 7; e^x = 6 =gt x = ln 6.`

Hence, the domain of function f(g(h(x))) comprises values fronm intervals `(0,ln6)U(ln 7,oo).`

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