Question

How do you find the domain and range for `y = sqrt(x^2 + 2x + 3)`?

Answer 1

Note that

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

`=(x+1)^2+2 >= 2 AA x in RR`

So `sqrt(x^2+2x+3)` is defined `AA x in RR` and the domain is the whole of `RR`.

Also we see that the range is `{y in RR: y >= 2}`. The minimum value `y=2` occurs when `(x+1) = 0`, i.e. when `x=-1`.