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A comprehensive overview of the concept of functions, including their definition, different representations (tables, graphs, verbal descriptions, and formulas), and various types of functions such as polynomials, rational functions, algebraic functions, and trigonometric functions. It also discusses operations on functions, including arithmetic operations and composition, as well as vertical and horizontal shifts and stretches of function graphs. The document aims to introduce the fundamental principles of functions and their applications, laying the groundwork for further study in mathematics and related fields. The detailed explanations, examples, and exercises make this document a valuable resource for students and learners seeking to deepen their understanding of this crucial mathematical concept.
Typology: High school final essays
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In everyday language the word “function” has at least two separate meanings I can think of: A. The purpose of something, as in “The function of a teacher is to impart knowledge.” B. When the value of some item somehow determines uniquely the value of some other item, as in “Your (alphabetized) last name determines uniquely your position on the (numbered) attendance sheet” (we say that the position is a function of the last name) or as in
“the radius of a sphere determines uniquely its volume” (we say the volume is a function of the radius.) In this course we will deal exclusively with the second interpretation: the value of some item (usually called ) somehow determines uniquely the value of some other item (usually called ) (we say is a function of and write ) With the wisdom of more than one century of thought we give the definition: Definition. A function is the following three things:
1 A table:
4. (The most common) An explicit formula Fun question: This is the volume of something, of what?
where D is a set of real numbers, R is a set of real numbers, and the function may be a graph, a formula or both. In fact, with few exceptions both D and R will be the entire set of real numbers, and we will spend a fair amount of time learning how to graph functions in cartesian coordinates and, conversely, to infer properties of a function from its graph.
III. Algebraic functions. Any function obtained by repeatedly and successively applying in any order any of the following algebraic operations: Things can get pretty wild with just these 5 simple operations! Here is an example:
IV. Trigonometric functions. The following six functions and algebraic combinations thereof. As usual, once again things can get pretty wild, you write some crazy expression involving and the above six functions! (Have some fun !)
There is another operation we can perform, with very useful results. It is called “composition” It is denoted by (note the little circle !) and it is defined by i.e., given , first compute , then apply the function to the result you got.
Pictorially the composition (first then ) is represented by This diagram makes clear that the values obtained by the first function must be part of the domain of the second function. It also makes clear that composition of functions is NOT commutative! (Putting on socks and putting on shoes do NOT commute !)
VERTICAL AND HORIZONTAL SHIFTS AND STRETCHES This composition operation, together with the old arithmetical ones, gives us a neat way to create new functions from old ones. In what follows figure out first what composition we are using. shifts the graph vertically (up if ) shifts the graph horizontally (left if )
stretches the graph vertically (enlarges if ) (Z if we reflect in first!) stretches the graph horizontally (compresses if ) (Z if we reflect in first!)
