What Is a Function?

A function tells you the relationship between two variables. You can look at a function as a number machine. You put a number into the function, and you get another one in return. You can specify a function in three different ways: As an expression, as table and as a graph.

An example of a function is

$$y=x+2$$

Here, $x$ is a number of your own choosing. That’s why $x$ is known as the independent variable. For each number you choose $x$ to be, $y$ becomes a different number. That’s why $y$ is called the dependent variable, and we say that $y$ is dependent on $x$. The definition of a function is that for each value of $x$, there’s only one value of $y$.

It’s also common to give the functions their own names. Because the word “function” begins with the letter “$f$”, you typically call your first function $f$. Directly after the name, we write the independent variable in parentheses. Since you’re mostly working with functions that are dependent on $x$, we set the function’s name to be $f(x)$ (which is read as “$f$ of $x$”). If there are several functions in the same exercise, each of them gets their own name. We have already used $f$, so we skip to the next letter in the alphabet and call the next function $g(x)$, and the next one $h(x)$, and so on. We can therefore write the function above as

$$f(x)=x+2$$