Range of $\sqrt{x-1}$

$\sqrt{z}$ is a function(Maps a point to a single point). So $\sqrt{z^2}$can only take a single value. By convention, $\sqrt{z}\ge0$. So,$\sqrt{z^2}=|z|$. So to answer your question, range will be $[0,\infty)$

We think of $\sqrt{}$ as the positive square root. This is for convenience, but it is the consensus of all mathematicians. It is of course true that $(-x)^2 = (x)^2$, so you might even say something like $x$ and $-x$ are both square roots of $x^2$. In solving an equation like $(x+4)^2 = 25$, you can only "take the square root" of both sides if you remember that there can always be up to two square roots, and you would want to write $(x+4) = \pm\sqrt{25}$.

But mathematicians agree that $\sqrt{}$ should always refer to one number, as functions must do. Confusion would ensue if when you wrote $\sqrt{}$ and when I wrote $\sqrt{}$, we might be referring to two separate numbers. So we want $\sqrt{}$ to be a function. Your question emphasizes that $\sqrt{}$ is a function by referring to it as $f(x)$ and asking about its range. Functions must output a $single$ value for each input. This is why we define $\sqrt{ x^2} = |x|$. The range is $[0,\infty)$.