# what is the largest domain of a function on a real line?

hey guys i have function which is ln(x)

the domain that i know is when X >0;

the question is State the domain of f. Make it the largest possible domain on the real line.

does that mean like the largest number on x or something. This question for me leads to a solving a taylor polynomial so help me out guys .

• I am wondering what the meaning of $ln(x)$ or $x\mapsto x^2$ should be if one hasn't specified the domain and codomain. – Curufin Mar 7 '12 at 12:00

You have described the domain informally but correctly. The largest possible domain of $\ln x$ is the set of all positive real numbers. Formally, one could write, for example, that the largest possible domain of $\ln x$ is $$\{x: (x\in \mathbb{R})\land (x>0)\}$$ (the set of all $x$ such that $x$ is a real number and $x>0$).
There are more informal ways to put it, but $x>0$ is probably too informal. You should make it clear that this largest possible domain is a set.
If we are working in the reals, the largest possible domain for the function $f(x)=x^2$ is $\mathbb{R}$, the set of all real numbers.
However, with this domain, the squaring function has an unfortunate flaw. It is not one-to-one (injective). There is no inverse function. The problem is that for example, $(-3)^2=3^2$, so knowing $f(x)$ does not allow us to recover $x$ uniquely. Thus we may wish to restrict the domain of $f(x)$ to the non-negative reals.