We will assume familiarity with functional notation. For example, $f(x)=x+1$ means $f(2)=3$.

In this chapter we take a step back to ensure that everything will be “well-behaved”.

This means specifying the $x$-values we are allowed to “plug” into a specific function (the domain) and ensuring that we get exactly one value $f(x)$ after doing that.

In particular, $f(x)$ cannot be undefined, nor can it take two or more different values.

A relation $f$ is a function if every element $x$ in its domain is mapped to exactly one value $f(x).$

The domain is the set of all possible inputs (”$x$“-values) a function takes.

We denote the domain of a function $f$ by $D_f.$

We define a function by specifying its rule and by giving a description of its domain.

Example

The function $f$ is defined by

This means that we can write it as $f(x)=x^2,$ plug in values like $f(-2)=4$ (as long as $x$ lies within the domain), and the domain of $f$ is given by $D_f = [-3, \infty).$