Answer
Domain: $[1, ∞)$
Range: $[3, ∞)$
Work Step by Step
Function: $f(x)=\sqrt {x-1} +3$
We can't have $x-1 < 0$, so $x\geq1$
We can have $x=1$, since we can take the square root of zero.
$f(x)= \sqrt {x-1} +3$
$f(1)= \sqrt {1-1} +3$
$f(1) = \sqrt 0 +3$
$f(1) = 0 +3$
$f(1) =3$
Since $x=1$ 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 $x$.
