Sections 3.1 and 3.2 Relations and Functions. A Refresher on Set Theory Experience says that a brief refresher on some basic notions is welcome, if not

Sections 3.1 and 3.2

Relations and Functions

A Refresher on Set Theory

Experience says that a brief refresher on some

basic notions is welcome, if not completely

necessary, at this stage. To that end, we present a

brief summary of ‘set theory’ and some of the

associated vocabulary and notations we use in the

text. Like all good Math books, we begin with a

definition.


Definition: A set is a well-defined collection of

objects which are called the ‘elements’ of the set.

Here, ‘well-defined’ means that it is possible to

determine if something belongs to the collection or

not, without prejudice.


For example, the collection of letters that make up

the word “smolko” is well-defined and is a set, but

the collection of the worst math teachers in the

world is not well-defined, and so is not a set.

In general, there are three ways to describe sets.

They are,

Ways to Describe Sets

1. The Verbal Method: Use a sentence to define a set.

2. The Roster Method: Begin with a left brace { , list each element of the set only once and then end with a right brace }.

3. The Set-Builder Method: A combination of the verbal and roster methods using a ‘dummy variable’ such as x.


For example, let S be the set described verbally as

the set of letters that make up the word ‘smolko’.

A roster description of S would be {s, m, o, l, k}.

Note that we listed ‘o’ only once, even though it

appears twice in ‘smolko’.

Also, the order of the elements does not matter, so

{k, l, m, o, s} is also a roster description of S.


A set-builder description of S is:

{x | x is a letter in the word ‘smolko’}

The way to read this is: The set of all elements x

such that x is a letter in the word ‘smolko’.

In each of the above cases, we may use the familiar

equals sign and write S {s, m, o, l, k} or

S {x | x is a letter in the word ‘smolko’}


S {x | x is a letter in the word ‘smolko’ }

{s, m, o, l, k}

Clearly m is in S and q is not in S. We express

these sentiments mathematically by writing m S

and q S.

Sets of Numbers

1. The Empty Set: {}{x| x x}. This is the set with no elements. Like the number ‘0’, it plays a vital role in mathematics.

2. The Natural Numbers: {1, 2, 3, …}.

3. The Integers: {…, 3, 2, 1, 0, 1, 2, 3, …}.

Sets of Numbers

4. The Rational Numbers:

{a/b | a and b and b 0}.

It turns out that another way to describe the rational numbers is:

{x | x possesses a repeating or terminating decimal representation}.

Sets of Numbers

5. The Real Numbers:

{x | x possesses a decimal representation}.

6. The Irrational Numbers:

{x | x is a non-rational real number}.

Said another way, an irrational number is a number whose decimal representation neither repeats nor terminates.

Sets of Numbers

7. The Complex Numbers:

{x | x a + bi and a , b and i 1 }.

Despite their importance, the complex numbers play only a minor role in the text.

Interval Notation

For the most part, this course focuses on sets

whose elements come from the real numbers .Recall that we may visualize as a line.

Segments of this line are called intervals of

numbers.

Below is a summary of the so-called interval

notation associated with given sets of numbers.

Interval Notation

Examples

As an example, consider the sets of real numbers

described below.

Intersection and Union of Sets

We will often have occasion to combine sets. There

are two basic ways to combine sets: intersection

and union. We define both of these concepts below.

Definition: Suppose A and B are sets. The intersection of A and B is defined to be the set

A B {x | x A and x B }. The union of A and B is defined to be the set

A B {x | x A or x B or both}.


Examples: If A [5 , 3) and B (1 , ), then we

can find A B and A B graphically.

To find A B , we shade the overlap of the two and

obtain A B (1 , 3).

To find A B, we shade each of A and B and

describe the resulting shaded region to find

A B [5 , ).


If A [5 , 3) and B (1 , ), then we can find

A B and A B graphically.

More Examples

Express the following sets using interval notation.

1. {x | x 2 or x 2}

2. {x | x 3}

3. {x | x 3}

4. {x | 1< x 3 or x 5}

Solutions

1. {x | x 2 or x 2}

Solutions

2. {x | x 3}

Solutions

3. {x | x 3}

Solutions

4. {x | 1< x 3 or x 5}

The Cartesian Coordinate Plane

In order to visualize the pure excitement that is

Pre-calculus, we need to unite Algebra and

Geometry. Simply put, we must find a way to draw

algebraic things. Let us start with possibly the

greatest mathematical achievement of all time:

The Cartesian Coordinate Plane.


Imagine two real number lines crossing at a right

angle at 0 as drawn below.


The horizontal number line is usually called the x-

axis while the vertical number line is usually

called the y-axis.

As with the usual number line, we imagine these

axes extending o indefinitely in both directions.


Having two number lines allows us to locate the

positions of points off of the number lines as well

as points on the lines themselves.

For example, consider the point P on the next slide.

To use the numbers on the axes to label this point,

we imagine dropping a vertical line from the x-axis

to P and extending a horizontal line from the y-axis

to P.


This process is sometimes called ‘projecting’ the

point P onto the x- (respectively y-) axis.


The projections of P onto the x and y axis are called

respectively, the x and the y coordinates of P.


This way P is represented by the ordered pair of

numbers x 2 and y 4. We write P (2 , 4).


In general, any point in the plane is an ordered pair

on real numbers. P (x , y).

P (x , y)


Therefore, the plane itself is the set of all possible

pairs P (x , y) of real numbers.

P (x , y)


The axes divide the plane into four regions called

quadrants. They are labeled with Roman numerals

and proceed counterclockwise around the plane:

Relations and Functions

Relations

In certain sense, all of Pre-calculus can be thought

of as studying sets of points in the plane.

With the Cartesian Plane now fresh in our memory

we can discuss those sets in more detail and as

usual, we begin with a definition.

Definition: A relation is a set of points, (ordered

pairs) in the plane.

Relations

Since relations are sets, we can describe them using

the techniques presented before. That is, we can

describe a relation verbally, using the roster

method, or using set-builder notation. Since the

elements in a relation are points in the plane, we

often try to describe the relation graphically or

algebraically as well.

Relations

Depending on the situation, one method may be

easier or more convenient to use than another.

As an example, consider the relation

R {(2 , 1), (4 , 3), (0 , 3 )}

As written, R is described using the roster method.

Since R consists of points in the plane, we follow

our instinct and plot the points.

Relations

Doing so produces the graph of R.

Examples of Relations

Graph the following relations.

Solutions

The graph for examples 1 and 2 are,

Solutions

The graph for example 3 is,

Solutions


Solutions


Solutions


Relations defined by Equations

In this section, we delve more deeply into the

connection between Algebra and Geometry by

focusing on graphing relations described by

equations. The main idea of this section is the

following.


The graph of an equation is the set of points

which satisfy the equation.

That is, a point (x , y) is on the graph of an equation

if and only if x and y satisfy the equation.

Notice that the graph of an equation is a set of

points in the plane and therefore by definition, a

relation


Here, “x and y satisfy the equation” means “x and y

make the equation true”.

It is at this point that we gain some insight into the

word “relation”.

If the equation to be graphed contains both x and y,

then the equation itself is what is relating the two

variables.

Examples

More specifically, in the next examples, we will

consider the graph of the equation x2 + y3 1.

Even though it is not specifically spelled out, what

we are doing is graphing the relation

R {(x , y) | x2 +y3 1}

Examples

The points (x , y) we graph belong to the relation R

and are necessarily related by the equation

x2 + y3 1, since it is those pairs of x and y which

make the equation true.

Determine whether or not (2 , 1) is on the graph of

x2 + y3 1.

Examples

To determine whether or not (2 , 1) is on the graph

of x2 + y3 1, we substitute x 2 and y 1 into

the equation to see if the equation is satisfied.

Hence, (2 , 1) is not on the graph of x2 + y3 1.

?2 3(2) ( 1) 1

3 1

Examples

We now graph the equation x2 + y3 1.

To efficiently generate points on the graph of this

equation, we first solve for y in terms of x2 3

3 2

33 23

3 2

1

1

1

1

x y

y x

y x

y x

Examples

We now substitute a value in for x, determine the

corresponding value y, and plot the resulting point

(x , y). For example, substituting x 3 into

the equation yields

so the point (3 , 2) is on the graph.

3 2 2 331 1 ( 3) 8 2y x

Examples

Continuing in this manner, we generate a table of

points which are on the graph of the equation. The

points are then plotted in the plane as shown below.

Examples

Remember, these points constitute only a small

sampling of the points on the graph of this

equation. To get a better idea of the shape of the

graph, we could plot more points until we feel

comfortable “connecting the dots”.

Doing so would result in a curve similar to the one

pictured below on the far left.

Examples

The graph x2 + y3 1 is given by

Intercepts of the graph

Definition: Suppose the graph of an equation is

given. A point on a graph which is also on the x-axis is called

an x-intercept of the graph. A point on a graph which is also on the y-axis is called

an y-intercept of the graph.

Intercepts of the graph

Finding the Intercepts of the Graph of an Equation

Given an equation involving x and y, we find the

intercepts of the graph as follows:

x-intercepts have the form (x , 0); set y 0 in the equation and solve for x.

y-intercepts have the form (0 , y); set x 0 in the equation and solve for y.

Domain and Range of a Relation

Suppose R is a relation. The sets of all x- and y-

coordinates of the points in R are given special

names which we define below.

The set of the x-coordinates of all the points in R is called the domain of R.

The set of the y-coordinates of all the points in R is called the range of R.

Examples of Domain and Range

Find the domain and range of the following relations

and express them in interval notation whenever

possible

Functions

Introduction to Functions

One of the core concepts in College Algebra is that

of function. There are many ways to describe a

function and we begin by defining a function as a

special kind of relation.

Definition: A relation F in which each x-coordinate

is matched with only one y-coordinate is said to

describe y as a function of x.


In other words, a function F is a relation with the

particular property that no two pairs of points in the

F have the same x-coordinate.

That is, in order to say y is a function of x, we just

need to ensure the same x-coordinate is not used in

more than one point.


Which of the following relations describe y as a

function of x?

A quick scan of the points in R1 reveals that the x-coordinate 1

is matched with two different y-coordinates: namely 3 and 4.

Hence in R1, y is not a function of x, or, R1 is not a function.


Which of the following relations describe y as a

function of x?

On the other hand, every x-coordinate in R2 occurs only once

which means each x-coordinate has only one corresponding

y-coordinate. So, R2 does represent y as a function of x, or, R2 is

a function.



In order to see what the concept of function means

geometrically, we graph R1 and R2 in the plane.


The fact that the x-coordinate 1 is matched with

two different y-coordinates

in R1 presents itself

graphically as the points

(1, 3) and (1, 4) lying on

the same vertical line, x 1.

Introduction to FunctionsIntroduction to Functions

If we turn our attention to the graph of R2, we see

that no two points of the

relation lie on the same

vertical line.

Introduction to FunctionsIntroduction to Functions

The Vertical Line Test: A set of points in the

plane represents y as a function of x if and only if

no two points lie on the same vertical line.

In other words, a relation R represents y as a

function of x if and only if no two points in R lie on

the same vertical line.

Vertical Line Test

Use the Vertical Line Test to determine which of

the following relations describes y as a function of

x.

Vertical Line Test

Use the Vertical Line Test to determine which of

the following relations describes y as a function of

x.

Vertical Line Test

Domain and Range of a Function

Suppose F is a relation that defines y as a function

of x. That is, F is a function. Then, as before we

have, The set of the x-coordinates of all the points in F is

called the domain of F. The set of the y-coordinates of all the points in F is

called the range of F.


Example: Find the domain and range of the

function

F {(3 , 2), (0 , 1), (4 , 2 ), (5 , 2 )}

Solution:

The domain of F is the set of the x- coordinates of

the points in F, namely, Dom F {3 , 0 , 4, 5}

and the range of F is the set of the y-coordinates,

Namely, Ran F {1 , 2}


Example: Find the domain and range of the

Function G whose graph is given below.


To determine the domain and range of G, we need

to determine which x and y values occur as

coordinates of points on the given graph. To find

the domain, it may be helpful to imagine collapsing

the curve to the x-axis and determining the portion

of the x-axis that gets covered. This is called

projecting the curve onto the x-axis.


Before we start projecting, we need to pay attention

to two subtle notations on the graph: the arrowhead

on the lower left corner of the graph indicates that

the graph continues to curve downwards to the left

forever more; and the open circle at (1, 3) indicates

that the point (1, 3) is not on the graph, but all

points on the curve leading up to that point are.


Dom G {x | x < 1} ( , 1)


Ran G {x | x 4} ( , 4]

Remark about Functions

All functions are relations, but not all relations

are functions. Thus the equations which described

the relations in previous examples may or may not

describe y as a function of x.

The algebraic representation of functions is the

most important way to view them so we need a

process for determining whether or not an equation

of a relation represents a function.

Three Examples

Determine which equations represent y as a

function of x.

1. x3 + y2 1

2. x2 + y3 1

3. x2y 1 3y

Solution to Example 1

1. x3 + y2 1Notice that in this case we are referring to the

relation defined by

R {(x , y) | x3 +y2 1}

We solve for y and determine whether each choice

of x will determine only one corresponding value

of y.


Solving for y in terms of x, we get3 2

2 3

2 3

3

3

1

1

1

1

1

x y

y x

y x

y x

y x


If we substitute x 0 into our equation for y, we

get y 1, so that (0 ,1) and (0 ,1) are on the

graph of this equation.

Hence, this equation does not represent y as a

function of x. Or equivalently, the relation

R {(x , y) | x3 +y2 1}

is not a function.


The graph of the relation

R {(x , y) | x3 +y2 1}

is given by


2. x2 + y3 1Notice that in this case we are referring to the

relation defined by

f {(x , y) | x2 + y3 1}



of y.


Solving for y in terms of x, we get

For every choice of x, the equation returns only one

value of y. Hence, this equation describes y as a

function of x.

2 3

3 2

3 2

1

1

1

x y

y x

y x


The graph of the function

f {(x , y) | x2 + y3 1}

is given by


3. x2y 1 3y

Notice that in this case we are referring to the

relation defined by

g {(x , y) | x2y 1 3y}



of y.


Solving for y we get

For every choice of x, the equation returns only one

value of y. Hence, this equation describes y as a

function of x.

2

2

2

1 3

3 1

1

3

x y y

x y

yx


The graph of the function

g {(x , y) | x2y 1 3y}

is given by

Function Notation

A function is a special kind of relation, one in

which each x-coordinate is matched with only one

y-coordinate. Here is another way of saying it:

A real-valued function f is a relation that assigns

to each real number x in a set X of numbers, a

unique real number y in a second set Y of numbers.

The set X is called the domain of the function f

and the second set Y is called the codomain of f.

Function Notation

For each element x in the domain X of the function,

the corresponding element y in Y is called the

image of x under the function f.

The image is denoted by f (x), that is, y = f (x).

f (x) is read “f of x.”

The set of all images of the elements of the domain

is called the range of the function.

DOMAIN RANGE

X Y

f

x

x

x

y

y

A way to picture a function is by an arrow diagram

Not in the range of f

Any symbol that represents an arbitrary number in the domain of a function f is called an independent variable. Any symbol that represents a number in the range of f is called a dependent variable.

A function, or relation in general, can be specified:

algebraically: by means of a formula numerically: by means of a table graphically: by means of a graph

Function Notation

The domain of a function is not always specified explicitly.

If no domain is specified for the function f, we take the domain to be the largest set of numbers x for which f (x) makes sense.

This "largest possible domain" is sometimes called the natural domain or implied domain.

Note on Domains

2( ) 3 2f x x is a function.

( )f x h

Example:

2(5) 3(5) 2 77f

23 2x h

2 23 6 3 2x xh h

Is a function represented by a formula. It has the format y f (x) “expression in x”

Substitute 5 for x

Substitute x+h for x

Algebraically Defined Function

2( ) 3 2f x x is a function. Example:

In this case the natural domain of the function is the set of all real numbers. That is,

Dom f (– , )

Is a function represented by a formula. It has the format y f (x) “expression in x”


4( )

1s t

t

is a function. Example:

In this case the natural domain of the function is the set

| 1 0 | 1Dom s t t t t

( ,1) (1, )Dom s

In interval notation this is


( ) 2 3h z z is a function. Example:

In this case the natural domain of the function consists of all values of z such that

2 3 0 or 3 2 or 2 / 3z z z

[ 2 / 3, )Dom h In interval notation this is


This is the case when we give numerical values for the function (the outputs, say the y-values) for certain values of the independent variable, say x.

In this case the function is represented by a table which looks like.

x-values x1 x2 … … xn

y = f (x) f (x1) f (x2) … … f (xn)

Numerically Specified Function

Notice that the table

defining the function f is the relation

f {(x1 , f (x1)), (x2 , f (x2)), … (xn , f (xn))}

written in row format, where the first row is the domain of f and the second row is the range of f.

x-values x1 x2 … … xn

y = f (x) f (x1) f (x2) … … f (xn)


x 0 1 2 3.7 4

f (x) 3.01 -1.03 2.22 0.01 1

Example: Suppose that the function f is specified by the following table.

Then, f (0) is the value of the function when x = 0. Thus

f (0) = 3.01 Look on the table where x = 0

f (1) = 1.03 Look on the table where x = 1

and so on


Example: The human population of the world P depends on the time t.

The table gives estimates of the worldpopulation P (t) at time t, for certainyears. For instance,

However, for each value of the time t,there is a corresponding value of P,and we say that P is a function of t.

(1950) 2,560,000,000P


Example: The human population of the world P depends on the time t.


t 0 1 2 3 4

V(t) 2.2 3.55 4.9 6.25 7.6

Example: The data represents the velocity V of an object, in feet/sec, after t seconds have elapsed.

Note: at 2 seconds the object is going at 4.9 ft/sec, that is V(2) = 4.9 ft/sec.

The table can be represented graphically as follows


-1 1 2 3 4 5 6 7 8

-1

1

2

3

4

5

6

7

8

t in seconds

V(t) ft/sec


Implicit Form of a Function

Implicit Form Explicit Form

F (x , y) constant y f (x)

Graphs of Functions

Obtaining Information from or About the Graph of a Function

The graph of a function is the set of all points (x, f (x)) in the xy-plane such that x is in the domain of f .

Sometimes the function is only known through its graph and may be very difficult to represent it algebraically. The next example illustrates this case.

In applications, the graph of a function often demonstrates more clearly the relationship between the independent variable x and the dependent variable y. Recall that,

Graphically Specified Function

The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. The figure shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994.

For a given value of t, the graph provides a corresponding value of a.


Example: The monthly revenue R from users logging on to your gaming site depends on the monthly access fee p you charge according to the formula

2( ) 5600 14000 0 2.5R p p p p

(R and p are in dollars.) Sketch the graph of R. Find the access fee that will result in the largest monthly revenue.


Solution: To sketch the graph of R by hand, we plot points of the form (p , R(p)) for several values of p in the domain [0 , 2.5] of R. First, we calculate several points.

2( ) 5600 14000 0 2.5R p p p p

p 0 0.5 1 1.5 2 2.5

R(p) 0 5600 8400 8400 5600 0


Graphing these points gives the graph in the figure on the left, suggesting the parabola shown on the right.


The revenue graph appears to reach its highest point when p = 1.25, so setting the access fee at $1.25 appears to result in the largest monthly revenue.


Example: The following table gives the weights, in pounds, of a particular child at various ages (in months) in her first year.

Age t 0 2 3 4 5 6 9 12

Weight W 8 9 13 14 16 17 18 19


Example: The following table gives the weights, in pounds, of a particular child at various ages (in months) in her first year.

Age t 0 2 3 4 5 6 9 12

Weight W 8 9 13 14 16 17 18 19


If we represent the data given in the table graphically by plotting the given pairs (t ,W(t)), we get, (connecting the successive points by line segments)

W(5) =

W(4.5)

16

Determine the domain, range, and intercepts of the function defined by the following graph.

4

0

-4

(0, -3)

(2, 3)

(4, 0)(1, 0) x

y

(10,0)

More Examples

More Examples

More Examples

Consider the function f (x) x /(x + 1).

1. Is the point (1 , 1/2) on the graph of f ?

2. If x 2, what is f (x) ? What point is on the graph of f ?

3. If f (x) 2, what is x ? What point is on the graph of f ?

Average Cost Function

The average cost function C of manufacturing x

computers per day is given by the function

1. Determine the average cost of manufacturing 30, 40, and 50 computers per day.

2. Graph the function C (x) for 0 < x 80 and find the value of x that minimizes the average cost

Finding Values of a Functionor

Evaluating a Function


Example: Let f (x) x2 + 3x + 4.

1. Find and simplify the following.

a) f (1), f (0), f (2)

b) f (2x), 2 f (x)

c) f (x + 2), f (x) + 2, f (x) + f (2)

d) (f (x + h) f (h)) /h where h 0

2. Solve f (x) = 4.

More Practice Problems

Is the Relation a Function?

Determine whether each relation represents a function.

If it is a function, state the domain and range.

1. {(2, 3), (4, 1), (3, -2), (2, -1)}

2. {(-2, 3), (4, 1), (3, -2), (2, -1)}

3. {(2, 3), (4, 3), (3, 3), (2, -1)}

Determine if the equation defines y as a function of x.

13

2y x

Determine if the equation defines y as a function of x.

22 1x y

Is the Relation a Function?


Example: Let f (x) 3x2 + 2x.

1. Find and simplify the following.

a) f (1), f (0), f (1)

b) f (2x), 2 f (x), f (x)

c) f (x + 1), f (x) + 1, f (x) + f (1)

d) (f (x + h) f (h)) /h where h 0

2. Solve f (x) = 5.

2

4

2 3

xf x

x x

2 9g x x

3 2h x x

Find the Domain of the Function

Find the domain of each of the following functions.

Write the answer in interval notation

The Algebra of Functionsor

Sum, Difference, Product,and Quotient of Functions

Function Arithmetic

Suppose f and g are functions and x is in both the

domain of f and the domain of g.

The sum of f and g, denoted f + g, is the function defined by the formula

( f + g)(x) f (x) + g (x) The difference of f and g, denoted f g, is the

function defined by the formula

( f g)(x) f (x) g(x)

Function Arithmetic

The sum of f and g, denoted f g, is the function defined by the formula

( f g)(x) f (x)g (x) The difference of f and g, denoted f g, is the

function defined by the formula

( f g)(x) f (x) g(x)

provided g(x) 0.

Function Arithmetic

Notice that in the previous definitions, x is in both,

the domain of f and the domain of g. That is, the

domain of f + g, f g, f g, and f g is the

intersection of the domain of f and the domain of g.

In addition, in the case of the quotient f g, the

values of x that make g(x) 0 must also be

excluded.

Examples

For f (x) 2x2 + 3 and g (x) x3 + 8 find f + g,

f g, f g, and f g and write their domains using

the interval notation.

( f + g)(x)

( f g)(x)

( f g)(x)

( f g)(x)

Examples

For f (x) 1/(x + 2) and g (x) x /(x 1) find f + g,

f g, f g, and f g and write their domains using the

interval notation.

( f + g)(x)

( f g)(x)

( f g)(x)

( f g)(x)

Documents

Sections 3.1 and 3.2 Relations and Functions. A Refresher on Set Theory Experience says that a brief refresher on some basic notions is welcome, if not