Question

What is the domain and range of `y = sqrt(x^2 + 2x + 3)`?

Answer 1

With radical functions the argument under the root-sign and the outcome are always non-negative (in real numbers).

Explanation:

Domain :
The argument under the root sign must be non-negative:
We 'translate' by completing the square:

`x^2+2x+3=(x^2+2x+1)+2=(x+1)^2+2`

Which is always `>=2` for every value of `x`

So there are no restrictions to `x`:
`x in(-oo,+oo)`

Range:
Since the lowest value the argument can take is `2`, the lowest value of `y=sqrt2`, so:
`y in [sqrt2,+oo)`