What is the domain of a function?

The domain of a function is all of the allowed input values of a function.

The domain can be defined in a number of different ways.

Here are some examples.

• Using set notation
  • $x=\{2,3,4,5\}$, this tells us that $x$, the input to the function, can be any of the numbers in the curly braces.
  • $x \in \mathbb{R}, x \neq -2 \hspace{1em}$ this means $x$, the input to the function, can be any real number except $-2$.
• Using inequalities
  • $-1 < x \leq 4$ where $x \in \mathbb{R}\hspace{1em}$ this means $x$, the input, can be any real number larger than $-1$ and less than or equal to $4$.
  • $x > 5$ this means $x$, can be any real number larger than $5$.
• Using words
  • $x$ can be any integer less than or equal to $-2$.

Sometimes we need to identify the domain from given information.

For example, if we are given the graph of a function, we can identify the domain by looking at the $x$-axis. The end-points of the function can either be a

• Filled circle $\rightarrow$ indicating that the end-point is included in the domain
• Open circle $\rightarrow$ indicating that the end-point is not included in the domain
• Arrow $\rightarrow$ indicating that the domain tends to infinity

If we are dealing with a rational (fractional) function then any $x$-values that make the denominator of the function equal zero must be excluded from the domain.

For example, if $f(x)=\dfrac{1}{x-3}$ then the domain could be $x \in \mathbb{R}, x \neq 3$, the domain restriction of $x \neq 3$ would be represented by a vertical asymptote with equation $x=3$.