Question

How do you find the domain and range of `f(x) = sqrt(x+6) /( 6+x)`?

Answer 1

Domain : `x!=-6` from the denominator.
`x>=-6` as an argument for a square root.

Explanation:

Together we get domain `x> -6`

Range:
Both the numerator and the denominator are now always positive.

So the range is `0 < f(x) < +oo`
graph{1/(sqrt(x+6)) [-10, 10, -5, 5]}
Note:
You may have noticed that `6+x=x+6=(sqrt(x+6))^2`, so under the given conditions we may rewrite:
`f(x)=cancel(sqrt(x+6))/(sqrt(x+6))^cancel2=1/(sqrt(x+6))`