For the given functions f and g find fog and state the domain of fog f(x)=sqrt x+9 g(x)=x2+4x-6.Need correct answer.

Expert Answers
hala718 eNotes educator| Certified Educator

`f(x)= sqrt(x + 9) `

`g(x)= x^2 + 4x -6`

`==gt fog(x)= f(g(x)) = f(x^2+4x-6)= sqrt(x^2+4x-6+9) = sqrt(x^2+4x+3)`

Now we will find the domain.

For the composite function we need to include the domain of the composite function and the input function.

==> First we will find the domain of `fog(x)= sqrt(x^2+4x+3)`

==> The domain are the real numbers such the function is not defined.

==> fog(x) is not defined when `x^2+4x+3 gt 0`

==> (x+3)(x+4) > 0

==> x >-3  and x>-4

`==gt x in (-3,oo)`

==> (x+3) < 0 and (x+4) < 0

`==gt x lt -3 and x lt -4 `

`==gt x in (-oo, -4) `

`==gtx in (-oo,-4)U(-3,oo)`

Also, we need to include the domain of the input function g(x).

The domain of g(x) is all real numbers.

==> Then, we conclude that the domain of fog(x) is the interval:

`x in (-00,-4)U(-3,oo).`