# 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 .

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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.

The largest possible domain of a function has nothing to do with the largest possible value (if there is such) that the function can take.

The following example may help with questions about largest possible domain. Look at the "squaring" function.

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.

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