A mathematical concept is well-defined (German wohldefiniert, French bien défini), if its content is independent on the form or the alternative representative which is used for defining it.

For example, in defining the power $x^r$ with $x$ a positive real and $r$ a rational number, we can freely choose the fraction form $\frac{m}{n}$ ($m\in\mathbb{Z}$ $n\in\mathbb{Z}_+$ of $r$ and take $$x^r := \sqrt[n]{x^m}$$ and be sure that the value of $x^r$ does not depend on that choice (this is justified in the entry fraction power). So, the $x^r$ is well-defined.

In many instances well-defined is a synonym for the formal definition of a function between sets. For example, the function $f(x) := x^2$ , is a well-defined function from the real numbers to the real numbers because every input, $x$ is assigned to precisely one output, $x^2$ However, $f(x) := \pm\sqrt{x}$ , is not well-defined in that one input $x$ can be assigned any one of two possible outputs, $\sqrt{x}$ or $-\sqrt{x}$

More subtle examples include expressions such as \begin{equation*} f\!\left(\frac{a}{b}\right) := a+b,\quad \frac{a}{b}\in\mathbb{Q}. \end{equation*}Certainly every input has an output, for instance, $f(1/2) = 3$ However, the expression is not well-defined since $1/2 = 2/4$ , yet $f(1/2) = 3$ , while $f(2/4) = 6$ , and $3 \neq 6$

One must question whether a function is well-defined whenever it is defined on a domain of equivalence classes in such a manner that each output is determined for a representative of each equivalence class. For example, the function $f(a/b) := a+b$ , was defined using the representative $a/b$ of the equivalence class of fractions equivalent to $a/b$