Sorry for the unclear title. Hopefully i'll clear it up here.
If for one $x$ value a part of my function becomes undefined, does that mean that the whole function is undefined at that part?
For example look at this function
$$f(x)=\ln(x^2+1)-\ln(x-1)$$
$f(0)$ makes the $\ln(1) = 0$, but the second one $\ln(-1)$ which we know is not solvable. Plotting the function on a graph makes the domain go from $1$ to $\infty$. Im kind of vexed here, because i've done functions like $f(x)=\frac{x-2}{x+2} +4x$ and this is function would still be defined for $f(-2)$.
Any help would be greatly appreciated.
it must be $$x\geq 1$$ since the definition of the logarithm function. In your second function we have $$x\ne -2$$ else we have a quotient with zero in the denominator