Answer
Domain: $[2,∞)$
Range: $[3,∞)$
Work Step by Step
Function: $f(x)= \sqrt {x-2}+3$
We can't have $x-2 < 0$; otherwise, we have a negative number under the square root radical. Thus, $x\gt2$.
We can have $x=2$, since we can take the square root of zero.
$f(x)= \sqrt {x-2} +3$
$f(2)= \sqrt {2-2} +3$
$f(2) = \sqrt 0 +3$
$f(2) = 0 +3$
$f(2) =3$
Since $x=2$ is the lowest value of the domain, we see that the lowest applicable value for the range is $f(x)=3$. The square root function has a positive coefficient, so the graph has increasing values for increasing values of $x$.
