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

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Answer 1 · 2016-04-25T20:23:44

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

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:
$y in [sqrt2,+oo)$

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