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DrDelMath

Intermediate Algebra
Elayn Martin-Gay

Chapter 3: Graphing

Section 3.6: Functions

Definition of Function

Several definitions are presented on Page 227 of the text -- relation, domain, range, and function. Because in this and the next course we study functions (not relations) I will combine the definitions in this textbook into the same definition of function which we use in College Algebra. We will not use the concept of relation except to get to the definition of function.

A set of ordered pairs is called a relation. The set of first coordinates is called the domain of the relation. The set of second coordinates is called the range of the relation. For example the set {(3,4), (2,8), (7,1), (7, 9), (5,5), (12,4)} is a relation. Its domain is the set of first coordinates {3, 2, 7, 5, 12} and its range is the set of second coordinates {4, 8, 1, 5, 9}.

Look carefully and you will observe that a relation consists of two sets (the domain and the range) and some kind of rule that pairs them up to produce the relation {(3,4), (2,8), (7,1), (7, 9),(5,5), (12,4)}.

This is not really careful mathematics but it has always helped me to understand what is going on. I think of arrows in each ordered pair which point from the first coordinate to the second coordinate -- those arrows constitute the rule.
{ ( 3,4 ),(2,8),(7,1),(7,9),(5,5),(12,4) }

Notice that an arrow emanates from each first coordinate. This is important!
By the way; this idea of an arrow to illustrate the association of a domain element with a range element will pervade many illustrations and explanations related to functions.
It is important to observe that the arrow only points from a domain element to a range element. The association only goes one way (contrary to everyday use of the word associate).
A domain element is associated with a range element -- a range element is not associated with a domain element.
A range element is the associate of a domain element -- a domain element is not the associate of a range element.

The textbook defines a function as a set of ordered pairs that assigns to each first coordinate exactly one second coordinate. Let's dissect this definition.

If a function is a set of ordered pairs then it follows that a function is a relation, but something (a restriction) has been added. The rule that assigns second coordinates to first coordinate has a severe restriction.

The phrase "assigns to each first coordinate" is the formal way of stating that an arrow emanates from each first coordinate. That is part of the definition of relation.

The words "exactly one second coordinate" is the formal way of stating that no domain element may have two elements emanating from it. This is the severe restriction. This is what distinguishes a function from a relation. This also gives rise to the "Vertical Line Test" which you will encounter shortly.

If we now focus only on the concept of function and put all this into one definition we obtain the following (which corresponds to the definintion used in our current College Algebra text at the top of Page 218)

Function: Definition and Notation

Picture of a Function

 

 

Function Notation

In Intermediate and College Algebra we direct most of our attention to the rule for a given function. Generally we take note of the domain only if ignoring it creates errors. In these courses. the range of a function rarely receives any attention, other than a few contrived exercises.

The rule for a function can be presented in a variety of ways, some of which are briefly presented here. Remember the rule of a function must be a rule which associates each domain element with a unique range element.

The rule for a function can be presented by simply listing the ordered pairs which constitute the function.
{(3,4), (2,8), (7,1), (7, 9),(5,5)}.

The rule of a function may be presented with arrows emanating at domain elements and terminating at range elements.
2 5 7 3 6 4 4 0

The rule of a function may be presented as a table.
Domain 3 28 9 5
Range 4 78 6 11

The rule of a function may be presented in words.
Suppose the domain is the set of Natural Numbers. The rule for a well known function states: "associate with each natural number the number of its positive divisors.
The natural number 5 has two divisors (1 and 5) so 5 is associated with 2.
The natural number 6 has 4 divisors (1, 2, 3, 6) so 6 is associated with 4.
Because each natural number has a unique number of divisors this simple little rule defines a function.

The rule for a function can be an equation. This is the most common way of describing the rule for a function. In Intermediate and College Algebra we will concentrate on functions whose rules are given as equations. Be careful not to fall into the trap of thinking the equation is the function or that a function is nothing more than an equation. The equations are nothing more than the rule of the function. The equation gives you the recipe for computing the range element when given a domain element.

A special notation has been created to facilitate using equations to specify the rule for a function. It is called function notation and is illustrated below.

Function notation is so extremely important, useful, and powerful throughout mathematics that it warrants considerable explanation. Function notation is an extremely compact notation which carries a lot of information. This is much more than introducing a variable. Function notation is read as: f of x, but if that is all it means to you, it will be worthless.

Observe that function notation takes the general form of something outside a set of parenthesis and something inside those parenthesis. In mathematics we name all our functions. The item outside the parenthesis is the name of the function. The item inside the parenthesis is a domain element. The entire collection (considered as a single symbol) of symbols f(x) is the unique range element associated with the domain element x by the function named f.
If you want to think in terms of arrows then this notation f(x) replaces xf (x ).

The name of a function is frequently a single letter but it need not be. We might name a function "phred" or "mq" or "area". In these cases we would speak of phred(x), mq(x), or area(x).

Picture of Function Machine Picture of Function Machine Until about 1980 the common example to illustrate the function concept was a hypothetical "function machine" as shown at the right. We spoke of input variables and output variables and in general were pretty sloppy with language about functions. Since then we have cleaned up our language. We now speak of domain elements and corresponding range elements and we write about them using function notation f(x). So today's hypothetical "function machine" would look more like the one pictured at the left.

We usually no longer use the above images to speak about hypothetical function machines because we now have real funciton machines -- the omnipresent calulators. Look at the keyboard shown below. The TI-83 is shown at the left and its keypad is at the left. Picture of Function Machine Picture of Function Machine

This (and others like it) is the ultimate function machine.
Find the following keys: ln, log, x2,x-1,sin, cos, tan.
On each of these keys, the label is the name of a mathematics function. When you enter a number into the display you are entering a domain element.
When you press a function key, you are asking the calculator to use the rule for the function you pressed to compute the unique range element corresponding to the domain element in the display.
After the calculator has completed its work it displays the unique range element corresponding to the domain element you had entered.

There can be no better illustration of the function concept.

Operation of those function keys is an exact physical implementation of the abstract concept of function.

Do not fall into the trap of calling f(x) the function. Remember f is the function and f(x) is the unique range element associated with x by the function named f.
Many mature mathematicians refer to f(x) as the function because of history and they have learned to compensate for the careless language. You will severely stunt your mathematical growth if you adopt such careless practices during the learning stage.

Example: Suppose the rule for a function is f(x)=3x+5.
When you see function notation you should remember every detail and word as defined in the schematic above. Then the rule will be correctly read as: The unique range element associated with the domain element x by the function f is 5 plus 3 times the domain element.
When speaking we read that rule as: "f of x equals three x plus five" with confidence that everyone involved in the conversation will understand that to mean: The unique range element associated with the domain element x by the function f is 5 plus 3 times the domain element.
Note that the = symbol may be translated as "is" or "equals" or "is equal to".

Example: Suppose the rule for a function is g(x)=x 2 2x+1.
When you see function notation you should remember every detail and word as defined in the schematic above. Then the rule will be correctly read as: The unique range element associated with the domain element x by the function g is the square of the domain element minus twice the domain element plus 1.
When speaking we read that rule as: "g(x) is equal to x squared minus two x plus 1" with confidence that everyone involved in the conversation will understand that to mean: The unique range element associated with the domain element x by the function g is the square of the domain element minus twice the domain element plus 1.
Note that the = symbol may be translated as "is" or "equals" or "is equal to".

Example: Suppose the rule for a function is g(x)=x 2 2x+1.
To calculate the value (or find) the unique range element associated with a particular domain element like 4, you substituted that domain element into the rule to obtain:
g(4)= 422(4 )+1=9. But you should never lose sight of the meaning of the notation so that you realize the unique range element associated with the domain element 4 by the function g is the square of the domain element 4 minus twice the domain element 4 plus 1.
Note that the = symbol may be translated as "is" or "equals" or "is equal to".

Whenever you see, hear, or say the word function you should retrieve the following diagram from memory and use it to clarify your thinking about the word function.

 

Picture of Function Notation

 

 

Equations can be either true of false.
Functions are neither true nor false - they simply exist.
If the rule for a function is an equation, then the graph of the function is simply the graph of its rule.
However, to extend the concept of graph to all functions we recognize that the graph of a function is the set of points of the form (a,f( a)).
Combined with function notation this definition of graph is very helpful.

A graph is not something you do. A graph of an equation or of a function is a mathematical object.

Write the definition of function.
Draw the Venn diagram used to visualize a function.
Write the standard function notation and by labeling the parts explain what each part means.
Function Notation Exercises and Solutions Supplemental Exercises:Part I: 5, 7, 12. Part II: 1, 4, 7 Solutions attached.
Page 23: Exercises:52, 60.
Page 458: Exercises:24, 30, 46.
Page 466: Exercises:23, 27, 36, 49.
Page 473: Exercises:77, 81, 82, 84.
Page 234: Exercises:3, 5, 8, 45, 52, 55, 63, 67
Page 854: Exercises:
Solve Exercises 10, 20, and 29. Be sure you can state the mathematics principle which permits you to perform each step in the process.
Sketch the graph of each of the equations in Exercises 10, 20, and 29.
Write the corresponding inequalities for the equations in Exercises 10, 20, and 29.
Use the first three properties of inequalities to solve the inequalities you associated with the equations in Exercises 10, 20, and 29.
Graph each inequality on its own line, then sketch the graph of the equation and both inequalities on the same line.
What observations can you make about this last graph?
Use the graph of the equation, results of testing one point, and deductive reasoning to solve the two inequalities you associated with the equations in Exercises 10, 20, and 29.
Graph each inequality on its own line, then sketch the graph of the equation and both inequalities on the same line.
What observations can you make about this last graph?

Solve Exercises 8, 21, and 25. Be sure you can state the mathematics principle which permits you to perform each step in the process.
Sketch the graph of each of the equations in Exercises 8, 21, and 25.
Write the corresponding inequalities for the equations in Exercises 8, 21, and 25.
Use the first three properties of inequalities to solve the inequalities you associated with the equations in Exercises 8, 21, and 25.
Graph each inequality on its own line, then sketch the graph of the equation and both inequalities on the same line.
What observations can you make about this last graph?
Use the graph of the equation, results of testing one point, and deductive reasoning to solve the two inequalities you associated with the equations in Exercises 8, 21, and 25.
Graph each inequality on its own line, then sketch the graph of the equation and both inequalities on the same line.
What observations can you make about this last graph?

Chapter 8: Functions and Graphs

Section 8.0: Cartesian Coordinate System

Coordinate Plane

Embedded in the following diagram are some obvious facts about the Cartesian Coordinate system, points in the system, and the Real Number coordinates of those points. Because these facts are obvious, the beginning student has a tendency to overlook them at times when their application might be appropriate. You are advised to study these facts and look for applications.
coordinate System

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The following two links provide useful insight into the distance formula.
The Distance Formula at PurpleMath distance formula link
The Distance Formula distance formula link

The following link provides a number of examples of the midpoint formula in action.
The Midpoint Formula at PurpleMath distance formula link
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Section 8.1A: Linear Equations in Two Variables - Review

The graph of an equation is a picture of its solution set.
Solutions of an equation in one variable are numbers. So the graph of an equation in one variable is a collection of points on the real number line.
Solutions of an equation in two variables are ordered pairs of numbers (x and y values. Better terminaology is: first coordinates and second coordinates) So the graph of an equation in two variables is a collection of points in The Cartesian Coordinate System.

Only two principles are needed to perform any manipulation of a linear equation. These two properties do not have an official name -- but many of us refer to them as "The First Two Properties of Equations."

These two simple properties have been used several times in previous topics about equations and will be used in all future discussions concerning equations of all kinds. So the smart thing to do is: "memorize them, remember them, review them regularly, and practice using them.

Recall that two equations are equivalent if they have the same solution set. In the past this was important when we were solving equations in one variable. The definition of equivalent equations does not change when we discuss equations in two variables. When we consider equations in two variables it is important to realize that every point on the graph of an equation is a solution of the equation and points not on the graph are not solutions.

When we manipulate an equation to improve our understanding of the graph of that equation it is important that we continue to view the same graph. This is why it is so important to always generate equivalent equations.

Page 191: Exercises: 1, 5, 21, 31.
Page 200: Exercises: 13, 20, 39,40.
Page 212: Exercises: 1, 5, 8.
Page 213: Exercises: 33, 39, 59, 63, 70.
Page 224: Exercises: 13, 15, 25, 27, 31, 34, 39 - 44, 67

Section 8.1B: Linear Functions

Page 516: Exercises: 3, 4, 13, 15, 18, 19, 21, 27, 35,43, .
Page 563: Exercises: 16, 17, 55.
Page 557: Exercises: 7, 9, 15.
Page 854: Exercises: 28, 31.
Page 466: Exercises: 40, 51.
Page 448: Exercises: 25, 54.