In what follows in this section, I’m going to give three concepts to help understand a function. One of them relies on the idea of a set in mathematics, so I’m going to very briefly describe that concept.
Definition2.1.1.
A set is any collection of things; the things in a set are called its elements. (In this course, sets will mostly be sets of numbers, but, in principle, they can be sets of anything.) If I want to write a set explicitly, I use braces: this is the notation for the set that contains the three numbers 4, 12 and 31.
\begin{equation*}
\{4, 12, 13\}
\end{equation*}
There is also a notation for membership. If I call the set I just defined \(A\text{,}\) then to show that 12 is a member, I write \(12 \in A\text{.}\) To show something is not a member, I use a crossed-out version of the same symbol: \(15 \notin A\)
Some of the most commonly used number sets have specific names an notations.
\(\NN\) is the set of natural numbers: positive whole numbers.
\(\ZZ\) is the integers: both positive and negative whole numbers.
\(\QQ\) is the rational numbers: number which can be written as fractions.
\(\RR\) is the real numbers: in addition to fraction, this include numbers which can’t be written as fractions, like \(\sqrt{2}\) or \(\pi\text{.}\)
Open inverals in the real numbers are written \((a,b)\text{.}\) This means all numbers between \(a\) and \(b\text{,}\) not including the endpoints.
Close inverals in the real numbers are written \([a,b]\text{.}\) This means all numbers between \(a\) and \(b\text{,}\) including the endpoints.
Subsection2.1.2Three Ideas for Functions
Calculus is a branch of mathematics concerned with the behaviour of functions. In this introductory/review section, I want to make sure to present a solid conceptual foundation of functions.
There are three major definitions or interpretations of functions.
First, a function is like a machine which acts on things (usually, but not necessarily, numbers). Functions are agents which perform certain actions. The function \(f(x) = x^2\) from \(\RR \rightarrow \RR\) is a process which takes a number and produces the square of that number. I can think of this \(f\) as the machine that squares numbers.
Second, a more abstract understanding of a function is as a rule that assigns something in one set to something in another. Let \(A\) and \(B\) be sets. A function \(f: A \rightarrow B\) is a rule that assigns an element of \(B\) to each element of \(A\text{.}\) The set \(A\) is called the domain. The range of the function is the subset of \(B\) consisting of all outputs of the function.
Finally, functions encode relationships and dependencies. For a function \(f: A \rightarrow B\text{,}\) I can thing of the elements of \(B\) depending on the elements of \(A\text{.}\) If I say that population growth is a function of food supply, I mean that there is a function which goes from numbers representing food supply to numbers representing growth rates. That function encodes the dependance of growth on food supply.
In addition to the three concepts, function are also visualized by graphs. To draw the graph of a function, I use the \(x\) axis as the input to the function (the set \(A\)) and the \(y\) axis as the output of the function (the set \(B\)). As you can see in Figure 2.1.2, from the point show on the graph of the quadratic, the vertical line drops down to the point on the \(x\) axis that represents the input, and the horizontal line goes across to the point on the \(y\) axis that represents the output. In addition to all the examples I show in Section 2.2, there is a library of functions in the common materials which shows the graphs of many frequently used functions.
Figure2.1.2.A Quadratic Function
Subsection2.1.3Functions on \(\RR\)
For the purposes of this course, I will present functions defined on \(\RR\) and its subsets. For most functions, I will not explicitly describe a domain; the domain of the function will implicitly be the largest subset of real numbers where the function applies. Likewise the range will implicitly be the subset of all possible outputs of the function.
Determining the domain of a function on \(\RR\) means avoiding undefined mathematical actions. There are three common restrictions.
I cannot divide by zero.
I cannot take even roots of negative numbers.
I cannot take logarithms of negative numbers or zero.
There are special domain restrictions for certain functions, such as inverse trig functions, but these three cover the vast majority of functions I will be presenting. Determining the domain of a function \(f\) means excluding real numbers which would lead to one of the three problems.
In addition, should I wish to, I can put additional domain restrictions on a function. Functions of time cannot extend infinitely back in time, so I will usually stipulate a starting time; the domain of the function will be after that starting time. Restricting domains is also useful to make a function invertible.
