For the given functions f and g, find fog and state the domain of fog. f(x)=10-sqrt x g(x)=x+12/7-x

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`f(x)= 10 - sqrtx `

`g(x)= (x+12)/(7-x)` 

`f(g(x)) = f((x+12)/(7-x)) = 10 - sqrt((x+12)/(7-x))`

`` Now we will find the domain.

We know that the domain are the values of x such that fog is defined.

First: `(7-x) shoule not be 0 ==gt 7-x != 0 ==gt x != 7` 

==> `x in (-oo,7)U(7,oo` )......(1)

Second: `(x+12)/(7-x)`  should be equal or greater than zero.

`==gt (x+12)/(7-x) gt= 0 `

`==gt (x+12) gt= 0 and (7-x) gt=0 `

`==gt x gt=-12 and x lt= 7 `

`==gt x in [-12,7` ]............(2)

Also,

`(x+12)lt= 0 and 7-x lt= 0 `

`==gt x lt= -12 and x gt= 7 `

`==gt x in phi.`

`` Then the domain is (1) and (2).

`==gt x in [ -12,7] nn (-oo,7) U (7,oo) `

`==gt x in [ -12,7] nn (-oo,7) U [-12,7] nn (7,oo) `

`==gt x in [-12,7) U phi ==gt x in [-12, 7)`

`` But the domain of f(x) is `[0,oo)`

`` => `x in [-12,7) nn [0,oo) `

`==gt x in [0, 7` )

Then, the domain of fog(x) is [0,7).

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