Question

How do you find the domain and range of `f(x)= sqrt(x+6)/(x-5)`?

Answer 1

Domain : the fraction and the root give limitations:

Explanation:

`x>=-6` or the argument under the square root will be negative.
`x!=5` or the denominator will be `=0`

Range : when `x=-6->f(x)=0`, but on the right side of the two-armed graph `x=0` is a horizontal asymtote.
`lim_(x-> oo) f(x)=0`
As `x` nears `5`, `f(x)` gets very large:
`lim_(x->5-) f(x)=-oo` and `lim_(x->5+) f(x)=+oo`

So there are no restrictions on the range.
graph{sqrt(x+6)/(x-5) [-5.04, 27.01, -8.67, 7.37]}