Question

What is the domain of `sqrt((x+7)/(3-x))`?

Answer 1

The domain is `x in [-7,3[`

Explanation:

Let `f(x)=(x+7)/(3-x)`

To determine the domain,

What`s under the `sqrt` sign is `>=0` with `x!=3`

So,

`f(x)>=0`

We build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-7` `color(white)(aaaaaa)` `3` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `x+7` `color(white)(aaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `3-x` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+` `color(white)(aa)` `||` `color(white)(aaaa)` `-`

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

Therefore,

`f(x)>=0` when `x in [-7,3[`