Mathematical Models

A mathematical model is an abstract concept through which we use mathematical language and notation to describe a phenomenon in the world around us. One example of a mathematical model is found in Dolbear’s Law, which has proven to be remarkably accurate for the behavior of snowy tree crickets. For even more of the story, including a reference to this phenomenon on the popular show The Big Bang Theory, see https://priceonomics.com/how-_to-_tell-_the-_temperature-_using-_crickets/. In the late 1800s, the physicist Amos Dolbear was listening to crickets chirp and noticed a pattern: how frequently the crickets chirped seemed to be connected to the outside temperature.

If we let represent the temperature in degrees Fahrenheit and the number of chirps per minute, we can summarize Dolbear’s observations in the following table.

For a mathematical model, we often seek an algebraic formula that captures observed behavior accurately and can be used to predict behavior not yet observed. For the data in the table above, we observe that each of the ordered pairs in the table make the equation

true. For instance, . Indeed, scientists who made many additional cricket chirp observations following Dolbear’s initial counts found that the formula above holds with remarkable accuracy for the snowy tree cricket in temperatures ranging from about F to F.

This model captures a pattern that is found in the world, and can be used to predict the temperature if only the number of chirps per minute is known. Not all phenomenon in the world that can be measured mathematically occur in a predictable pattern. In this section, we will study functions which are mathematical ways of formally studying situations where for a given input, such as the number of chirps above, there is one consistant output. For situations where a given input might give a variety of outputs, we encourage you to study statistics! Also note that this relationship is not causal. Even though the number of chirps is considered out “input” the increase in chirps does not cause the temperature to increase.

Functions

The mathematical concept of a function is one of the most central ideas in all of mathematics, in part since functions provide an important tool for representing and explaining patterns. At its core, a function is a repeatable process that takes a collection of input values and generates a corresponding collection of output values with the property that if we use a particular single input, the process always produces exactly the same single output.

For instance, Dolbear’s Law provides a process that takes a given number of chirps between and per minute and reliably produces the corresponding temperature that corresponds to the number of chirps, and thus this equation generates a function. We often give functions shorthand names; using “” for the “Dolbear” function, we can represent the process of taking inputs (observed chirp rates) to outputs (corresponding temperatures) using arrows:

Alternatively, for the relationship “” we can also use the equivalent notation “” to indicate that Dolbear’s Law takes an input of chirps per minute and produces a corresponding output of degrees Fahrenheit. More generally, we write “” to indicate that a certain temperature, , is determined by a given number of chirps per minute, , according to the process .

We will define a function informally and formally. The informal definition corresponds to the way we will most often think of functions, as a process with inputs and outputs.

The formal definition of a function will establish a function as a special type of relation. Recall that a relation is a collection of points of the form . If the point is in the relation, then we say and are related.

How is this definition consistent with the informal definition, which describes a function as a process? Well, if you have a collection of ordered pairs , you can choose to view the left number as an input, and the right value as the output. For a function , is defined as the unique value that is paired with. If does not appear among the first coordinates of the ordered pairs, then is not a possible input to , since there is no way to determine .

Domain and Range

We now give some important definitions that allow us to talk about the inputs and outputs of functions.

Sometimes, when we are given a function as a formula, we are not told the domain. In these circumstances we use the implied domain.

It is important to note that the definition of a function includes its domain and range. A function needs a rule and a domain to precisely determine a set of points, the range.

Interval Notation

Intervals (contiguous sections of the number line) play an important role in expressing the domains of many types of functions. As a standard way of writing these solutions, we rely on interval notation. Interval notation is a short-hand way of representing the intervals as they appear when sketched on a number line. As an example, consider which, when sketched on a number line, is given by

This sketch consists of a single interval with left-hand endpoint at and no right-hand endpoint (it keeps going). In interval notation, this would be written as . This is an example of a closed infinite interval, “closed” because the point at (the only endpoint) is included and “infinite” because it has infinite width. The solid dot at indicates that the point is included in the interval.

There are four different types of infinite intervals, two are closed infinite intervals (which contain their respective endpoint) and the other two are open infinite intervals (which do not contain the endpoint). For a fixed real number, these are:

(a)
represents ,
(b)
represents ,
(c)
represents , and
(d)
represents .

The notation indicates uses the square bracket to indicate that the endpoint is included and the round parenthesis to indicate that the endpoint is not included.

Not every interval is infinite, however. Consider the interval in the following sketch

which consists of all with . It is not an infinite interval, having endpoints at and . The endpoint at is not included, but the endpoint at is included. In interval notation this would be written as . As with the infinite intervals, the square bracket indicates that the right-hand endpoint is included and the round parenthesis endicates that the left-hand endpoint is not included. (This is an example of a “half-open interval”.)

For bounded intervals (ones that are not infinite), there are also four possibilities. For and both fixed real numbers, these are:

(a)
represents ,
(b)
represents ,
(c)
represents and
(d)
represents .

Practically, this amounts to writing the left-hand endpoint, the right-hand endpoint, then indicating which endpoints are included in the interval. When neither endpoint is included, can be mistaken for a point on a graph. You will need to use the context to know which is meant.

Intercepts

Zero is a very important number, and as we will see later, knowing where a function’s - or -value equals zero can be powerful information.