3.2 What is a function?

We can generalize these ideas to the statement ``how does a change in one variable impact the value of another''?

Mathematically, we describe relationships, through the use of an object called a function.

A function is simply a ``rule''. The rule takes a value as an input and provides a specific output.

Both the input and output are called variables, with the input often described as the independent variable and the output as the dependent variable.

Another common terminology is to call the input the x-variable and the output the y-variable.

The potential values that x can take are called the domain of the function.

The potential values that y can take are called the range of the function.

For the consumer loyalty example, the x-variable is consumer loyalty and the y-variable is the chance that they repeat purchase. The domain of the function that relates loyalty to chance of a repeat purchase is the interval [0,1] because that's how loyalty was defined, and the range of the function is again [0,1] but this time because it is a probability.

We can consider a function as depicted by the following diagram; an input goes in, the rule is in the box, and the output comes out. Pictorially it would look like this:

One of the benefits of thinking mathematically is that it provides a concise shorthand notation: in particular we represent a function in the form y = f(x).

Here, y is the output variable, x is the input variable and the rule is denoted simply by the letter f. In English, the expression y = f(x), can be translated as

The function f, takes an input variable, x, and produces an output variable, y.

For the marketing example described previously, the input is loyalty, the output is propensity to repeat purchase, and the function f describes how propensity depends on loyalty.