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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 12: SETS, RELATIONS AND FUNCTIONS


SECTION 12.1: SETS

set of animals

A "set" is a collection of "objects." Each object in the set is called an "element of the set." The elements usually have something in common.

For example, the following, {monkey, bear, lion, giraffe, zebra}, is called a set of elements. What do these elements have in common? One answer might be that they are animals in a local zoo.

Notice the symbols "{" and "}". We will call theses symbols "set enclosures."   "{" is the "left set enclosure" and "}" is the "right set enclosure."

The following set {2, 4, 6, 8, 10, ...} is a collection of objects that are numbers. The elements are 2, 4, 6, 8 and 10, and what these elements have in common is that they are positive even numbers. Notice that the set {2, 4, 6, 8, 10, ...} has 3 periods after the number 10. These three periods, called "ellipsis," mean “and so forth” or suggest that the elements in a set continue on forever or as mathematicians say "continue on to infinity."

When we need to represent some "recognizable pattern" mathematicans often use ellipsis. Consider the following set: {3, 6, 9,..., 90, 93, 96}. I have left out many numbers from this set but the ellipses suggest I am incrementing each subsequent number by 3 and the set contains all the numbers incremented by 3, starting at 3, going to the number 96.

Can you see the pattern in the following? {...1, 2, 4, 8, 16, 32, 64, 128...} If you take any number in this set and multiply it by 2 you will get the next number in the set. Pattern recognition skills are very important in the world of mathematics!

 

SECTION 12.2: DOMAIN, RANGE, RELATION

Let's look at this set: { (1,3), (-2,4), (5,-2), (-1,-3) } This is a set of ordered pairs which can be used to represent points in the Cartesian Coordinate System. The points below can be represented by our set of ordered pairs { (1,3), (-2,4), (5,-2), (-1,-3) }.

graph1

The set of x-values of a set of ordered pairs { (1,3), (-2,4), (5,-2), (-1,-3) } is named the "domain." The domain in this instance is {-2, 1, -1, 5}.

The set of y-values of a set of ordered pairs is named the "range." The range in this instance is {4, 3, -2, -3}.

The set of ordered pairs is also named a "relation." So { (1,3), (-2,4), (5,-2), (-1,-3) } is a relation.

EXAMPLES

ex. 1) State the domain and range of the relation {(5, –3), (4, 6), (3, –1), (6, 6), (5, 3)}. 

ans: The domain is {5, 4, 3, 6} and the range is {–3, 6, –1, 3}
Comment: Notice that the x coordinate, 5, occurs twice in {(5, –3), (4, 6), (3, –1), (6, 6), (5, 3)}. Mathematicians have agreed to write an x value in the domain once even though it may occur more than once in the relation. The same goes for the range: notice the y coordinate 6 occurs twice in the relation but is written only once in the range.

ex. 2) State the domain and range of the relation {(–3, 4), (–1, 4), (0, 4), (1, 4), (2, 4)}.

ans: The domain is {-3, -1, 0, 1, 2} and the range is {4}



SECTION 12.3: FUNCTIONS

There is a very special "relation" called a "function." Functions are very powerful in the world of applications as we will see in Chapter 16 of this manual.

WHAT IS A FUNCTION?

If each x-value of a relation has one unique y-value assigned to it, then the relation is named a "function."


ex. 1) { (1,3), (1, 5), (-2,4), (5,-2), (-1,-3) } is not a function! Look at the set of ordered pairs to your left. Note the ordered pairs in blue. The x-value 1 has two different y-values, 3 and 5.
Hence, we say { (1,3), (1, 5), (-2,4), (5,-2), (-1,-3) } is not a function.

ex. 2) { (1,3), (-2,4), (5,-2), (-1,-3) } is called a function.
Why? Because each x-value has only 1 y value.

ex. 3) { (7,5), (-7, 5), (-7,4), (-7,-2), (6,-3) } is not a function. Look at the set of ordered pairs to your left. The x-value -7 has three different y-values 5,4,and -2.
Hence, we say { (7,5), (-7, 5), (-7,4), (-7,-2), (6,-3) } is not a function.

ex. 4) { (7,-2), (-2,-2), (4,-2), (-3,-2) } is a function.
Why? Because each unique x-value has only 1 y-value.

COMMENT: All functions are relations but not all relations are functions. Remember, a relation is nothing more than a set of ordered pairs.


 

12.4 GRAPHS OF FUNCTIONS VERSES NON-FUNCTIONS

A set of ordered pairs that is a function will have more than one point falling on a vertical line. Let's compare the graph of a function to a graph of a non function.

FUNCTION   NOT A FUNCTION
{ (1,3), (-2,4), (5,-2), (-1,-3) }   { (1,3), (1, 5), (-2,4), (5,-2), (-1,-3) }
GRAPH2   GRAPH 3

 

FUNCTION   NOT A FUNCTION
{ (7,-2), (-2,-2), (4,-2), (-3,-2) }   { (7,5), (-7, 5), (-7,4), (-7,-2), (6,-3) }
graph 4   graph 5

We will be looking at functions in greater depth in chapter 16. This was a brief introduction.

 

 

-----------------------------
THE SUMMER STUDY GUIDE
BY CHAPTERS

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RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE

- CHAPTER 1: THE NUMBER SYSTEM

- CHAPTER 2: ORDER OF OPERATIONS

- CHAPTER 3: VARIABLES, MONOMIALS, BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS

- CHAPTER 4: SIGNED NUMBERS, ABSOLUTE VALUE, AND INEQUALITY SYMBOLS

- CHAPTER 5: FACTORS, COMMON FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS

- CHAPTER 6: PROPERTIES OF NUMBERS

- CHAPTER 7: THE WORLD OF FRACTIONS

- CHAPTER 8: EXPONENTS

- CHAPTER 9: ROOTS

- CHAPTER 10: ALGEBRAIC EXPRESSIONS

- CHAPTER 11: CARTESIAN COORDINATE SYSTEM

- CHAPTER 12: SETS, RELATIONS AND FUNCTIONS

- CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS

- CHAPTER 14: X-INTERCEPT(ZERO) AND Y INTERCEPT(B)

- CHAPTER 15: LINES

- CHAPTER 16: FUNCTIONS

- CHAPTER 17: MULTIPLYING POLYNOMIALS

- CHAPTER 18: FACTORING

- CHAPTER 19: RATIONAL EXPRESSIONS

- CHAPTER 20: SOLVING EQUATIONS

- CHAPTER 21:SOLVING INEQUALITIES

- CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS

- CHAPTER 23: QUADRATICS

- CHAPTER 24: CIRCLES

- CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES

- CHAPTER 26: VOLUMES