Math Functions Domain


A domain is a set of all inputs over which the function has defined outputs.



The domain contains all values which are accepted as the input of a function.

Keep following rules in mind when defining the domain :

  • Dividing by zero is not allowed!
  • It is not possible to take the root of a negative value.


Given is following function $f(x)=\frac1{x}$. Remember, it is not possible to divide a number by zero. For this value the function is not defined. In this case the domain is:

$\mathbb{R} \backslash \{0\}$

or a different notation would be




Both notations mean the same.

$\mathbb{R}$ stands for the real numbers. In other words, those are all the numbers that you know.


We can specify the domain in two different ways:

  • Set notation
  • Interval notation

Set notation

The previous two notations were examples for the set notation:

  • The set of all real numbers $\mathbb{R}$
  • The set of real numbers except zero: $\mathbb{R} \backslash \{0\}$
  • The set of all real numbers which are not equal to zero: $\{x\in\mathbb{R}|x\neq0\}$

Interval notation

An interval notation is also an option. All numbers in the interval are allowed.

  • All numbers between 0 and 3: $[0,3]$
  • All numbers between 0 and 3, except 0 (and 3 included): $]0,3]$


Closed square brackets mean that the number is included in the interval. Open square brackets mean that the number itself is not included in the interval anymore.