How to apply Newton's method to solve for a root of a function that is not everywhere defined?

Question

If I want to solve for a root of a nonlinear function $f$, I would naturally consider using Newton's method, starting from an initial guess $x_0$:

$$ x_{k+1} = x_{k} - \frac{f(x_k)}{f'(x_k)} $$

Now, if $f$ is not defined everywhere on $\mathbb{R}$, what can I do to make this work?

Examples: $f(x)=\sqrt{x}-1$ and $f(x) = \ln{x}$ are not defined for $x<0$ (both have a root at $x=1$)

During iteration, the Newton's method iterations could accidentally end up at a point $x_{k+1} < 0$. How to proceed from there?

Questions

1. Can Newton's method be adapted to take care of these situations, and how?
2. What alternative approaches are there?

Approaches that also work for the multi-dimensional case are encouraged.