That's tricky in this example, but more common in physics. Or you start with a simple model of the world and see what functions come out of that. Here you basically start with a function that you think is logical. That's when you might want to think of a theory to explain the function.

Even then you often have to make a judgement call about whether you think the function should be polynomial (e.g. For non-linear functions it's more tricky, but there are techniques. For linear functions you can use linear regression to find the equation of the line that best fits the data. Then you could plot the number of words over time on a graph and try to fit a function. You get a large sample of people, teach them some words and then test their ability to recall then after certain time periods. The first empirical, which means you do an experiment. There are two main ways that science comes up with models/functions. I'm almost certain that the modelling function in this question was made up purely for this question.

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