Question

How do you find the domain and range of `sqrt [x/(x-6)]`?

Answer 1

The domain is `x in (-oo,0]uu(6,+oo)`
The range is `[0,1)uu(1,+oo)`

Explanation:

To find the domain, what's under the square root sign is `>=0`

So,

`x/(x-6)>=0`

Let `p(x)=x/(x-6)`

We need to build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `0` `color(white)(aaaaaaaa)` `6` `color(white)(aaaaaaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-6` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `p(x)` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `||` `color(white)(aaaa)` `+`

Therefore,

`p(x)>=0` when `x in (-oo,0]uu(6,+oo)`

The domain is `x in (-oo,0]uu(6,+oo)`

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

When `x=0`, `p(x)=0`

`lim_(x->6^+)p(x)=+oo`

So,

the range is `[0,1)uu(1,+oo)`

graph{sqrt(x/(x-6)) [-25.67, 25.65, -12.83, 12.84]}