Question

How do you find the domain and range of `f(x) = sqrt{(x - 1)/(x + 4)}`?

Answer 1

The domain is `D_f(x)=x in ]-oo,-4[ uu [1,+oo[`
The range is `R_f(x)=[0, +oo[`

Explanation:

As you cannot divide by `0`, `x!=-4`

and what's under the `sqrt` sign is `>=0`

`(x-1)/(x+4)>=0`

Let `g(x)=(x-1)/(x+4)`

We build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaaa)` `-4` `color(white)(aaaaa)` `1` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+4` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-1` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `g(x)` `color(white)(aaaaaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aa)` `-` `color(white)(aaaa)` `+`

Therefore,

`g(x)>=0` when `x in ]-oo,-4[ uu [1,+oo[`

The domain is `D_f(x)=x in ]-oo,-4[ uu [1,+oo[`

`lim_(x->+-oo)g(x)=lim_(x->+-oo)x/x=lim_(x->+-oo)1=1`

`lim_(x->-4^-)g(x)=lim_(x->-4^-)-5/(0^-) = +oo`

`g(1)=0/5=0`

So,

the range is `R_f(x)=[0, +oo[`