Outline

Real Numbers and the Real Line

Sets
Intervals
Solving Inequalities
Absolute Value
Cartesian Coordinate System
Line Equations
Parallel and Perpendicular Lines
Distance and Circles in the Plane
Functions

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Preliminaries - Real Numbers and the Real Line
Much of calculus is based on properties of the real number system. Real
numbers are numbers that can be expressed as decimals, such as:

3
− = −0.75
4
1
= 0.3333 . . .
√3
2 = 1.4142 . . .

The real numbers can be represented geometrically as points on a

number line called the real line.

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Preliminaries - Real Numbers and the Real Line
Much of calculus is based on properties of the real number system. Real
numbers are numbers that can be expressed as decimals, such as:

3
− = −0.75
4
1
= 0.3333 . . .
√3
2 = 1.4142 . . .

The real numbers can be represented geometrically as points on a

number line called the real line.

The algebraic properties say that the real numbers can be added,
subtracted, multiplied, and divided (except by 0) to produce more real
numbers under the usual rules of arithmetic. You can never divide by 0!
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Preliminaries - Real Numbers and the Real Line

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Preliminaries - Real Numbers and the Real Line

We distinguish three special subsets of real numbers.

1. The natural numbers, namely 1, 2, 3, . . .

2. The integers, namely 0, ±1, ±2, ±3, . . .

3. The rational numbers, namely the numbers that can be expressed in

m
the form of a fraction , where m and n are integers and n , 0.
n
Examples are

1 4 −4 4 100
, − = = , 100 =
3 9 9 −9 1

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Preliminaries - Real Numbers and the Real Line

The rational numbers are precisely the real numbers with decimal
expansions that are either
terminating (ending in an infinite string of zeros), for example

3
= 0.75000 · · · = 0.75
4
eventually repeating (ending with a block of digits that repeats over
and over), for example

23
= 2.090909 · · · = 2.09
11
Real numbers that√are√not rational are called irrational numbers.
Examples are π, 4, 5 and log10 3.
3

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Preliminaries - Sets

A set is a collection of objects, and these objects are the elements of

the set.

If S is a set, the notation a ∈ S means that a is an element of S, and

a < S means that a is not an element of S.

If S and T are sets, then S ∪ T is their union and consists of all

elements belonging either to S or T (or to both S and T). The
intersection S ∩ T consists of all elements belonging to both S and T.

The empty set is the set that contains no elements.

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Preliminaries - Sets

Some sets can be described by listing their elements in braces. For

instance, the set A consisting of the natural numbers (or positive integers)
less than 6 can be expressed as

A = {1, 2, 3, 4, 5}.

Another way to describe a set is to enclose in braces a rule that generates

all the elements of the set. For instance, the set

A = {x | x is an integer and 0 < x < 6}

is the set of positive integers less than 6.

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Preliminaries - Intervals

A subset of the real line is called an interval if it contains at least two

numbers and contains all the real numbers lying between any two of
its elements.

Geometrically, intervals correspond to rays and line segments on the

real line, along with the real line itself. Intervals of numbers
corresponding to line segments are finite intervals; intervals
corresponding to rays and the real line are infinite intervals.

A finite interval is said to be closed if it contains both of its endpoints,

half-open if it contains one endpoint but not the other, and open if it
contains neither endpoint.

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Preliminaries - Types of Intervals

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Preliminaries - Solving Inequalities
The process of finding the interval or intervals of numbers that satisfy an
inequality in x is called solving the inequality.
Example
Solve the following inequalities and show their solution sets on the real
line.
x 6
a ) 2x − 1 < x + 3 b ) − < 2x + 1 c) ≥5
3 x −1

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Preliminaries - Solving Inequalities

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Preliminaries - Solving Inequalities

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Preliminaries - Absolute Value
The absolute value of a number x, denoted by |x | is defined by the
formula 
x

 x≥0
|x | = 
−x x < 0.


Geometrically, the absolute value of x is the distance from x to 0 on

the real number line. Since distances are always positive or 0, we see
that |x | ≥ 0 for every real number x, and |x | = 0 if and only if x = 0.
Also, on the real line

|x − y | = the distance between x and y .

√
Since the symbol a always denotes the nonnegative square root of
a, an alternate definition of |x | is
√
|x | = x 2.
√
It is important to remember that a 2 = |a |. Do not write unless you
already know that a ≥ 0.
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Preliminaries - Absolute Value

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Preliminaries - Absolute Value

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Preliminaries - Solving an Equation with Absolute Values

Example
Solve the equation |2x − 3| = 7.

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Preliminaries - Solving an Equation with Absolute Values
Example
2
Solve the inequality 5 − < 1.
x

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Preliminaries - Solving an Equation with Absolute Values
Example
Solve the inequality and show the solution set on the real line:
a) |2x − 3| ≤ 1 b) |2x − 3| ≥ 1.

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Preliminaries - Solving an Equation with Absolute Values

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Cartesian Coordinate System
.
. To begin, we draw two
perpendicular coordinate lines
that intersect at 0-points of
each lines. These lines are
called coordinate axes: The
horizantal line is called x-axes
and the vertical line is called
y-axes.

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Cartesian Coordinate System
.
. To begin, we draw two
perpendicular coordinate lines
that intersect at 0-points of
each lines. These lines are
called coordinate axes: The
horizantal line is called x-axes
and the vertical line is called
y-axes.

Points in the plane can be

identified with ordered pair of
real numbers.
P ↔ (a , b )

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Cartesian Coordinate System
.
. To begin, we draw two
perpendicular coordinate lines
that intersect at 0-points of
each lines. These lines are
called coordinate axes: The
horizantal line is called x-axes
and the vertical line is called
y-axes.

Points in the plane can be

identified with ordered pair of
real numbers.
P ↔ (a , b )

This coordinate system is called the rectangular coordinate system or

Cartesian coordinate system.
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Line Equations

.
Let P1 (x1 , y1 ) and P2 (x2 , y2 ) be
P2 (x2 , y2 )
two points on the plane. Then,
there is a unique line which
contain these points. The slope
∆ y = y1 − y2
of this line is
θ ∆y y2 − y1
P1 (x1 , y1 ) m= =
∆x x2 − x1
∆x = x1 − x2

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Line Equations

Definition (Slope)
The slope m of a line is defined as the change in the y coordinate divided
by the corresponding change in the x coordinate, between two distinct
points on the line.

∆y vertical change
m= =
∆x horizontal change

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The slope of a vertical line is undefined, since ∆x is zero for a vertical
line.
The slope of a horizontal is zero, since ∆y is zero for a horizontal line.

Horizontal line has slope 0

Vertical line has no slope

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Line Equations
The equation of a line passing through the point P (x0 , y0 ) and has slope
m is
y − y0 = m(x − x0 )

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Line Equations
The equation of a line passing through the point P (x0 , y0 ) and has slope
m is
y − y0 = m(x − x0 )

The equation of a horizontal line passing through the point P (x0 , y0 ) is

y = y0 (since the slope of a horizontal line is zero).

The equation of a vertical line passing through the point P (x0 , y0 ) is

x = x0 .

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Line Equations
The equation of a line passing through the point P (x0 , y0 ) and has slope
m is
y − y0 = m(x − x0 )

The equation of a horizontal line passing through the point P (x0 , y0 ) is

y = y0 (since the slope of a horizontal line is zero).

The equation of a vertical line passing through the point P (x0 , y0 ) is

x = x0 .

Note: The slope of the line defined by the linear equation ax + by + c = 0

is
a
m=−
b

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Line Equations

Example: Find the equation of the following lines.

1. the line passing through the points P (1, 2) and Q (5, 3).
2. the horizontal line passing through the point P (2, −1).
3. the vertical line passing through the point P (2, −1).

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Line Equations

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Parallel and Perpendicular Lines

Let L1 and L2 be two lines with slopes m1 and m2 , respectively. Then, we

say that
L1 and L2 is parallel (L1 //L2 ) if m1 = m2 .
L1 and L2 is perpendicular (L1 ⊥ L2 ) if m1 .m2 = −1.

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Parallel and Perpendicular Lines

Example: Find the equation of the line passing through P (−1, 2) and
parallel
perpendicular
to the line 2x + y = 4.

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Distance and Circles in the Plane
Distance Formula for Points in the Plane
The distance between P (x1 , y1 ) and Q (x2 , y2 ) is
q
d = (x1 − x2 )2 + (y1 − y2 )2 .

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Distance and Circles in the Plane
Distance Formula for Points in the Plane
The distance between P (x1 , y1 ) and Q (x2 , y2 ) is
q
d = (x1 − x2 )2 + (y1 − y2 )2 .

Example
Calculate the distance between the points P (−1, 2) and Q (3, 4).

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Distance and Circles in the Plane

Example
Find the distance from origin to the point P (x , y ).

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Distance and Circles in the Plane

By definition, a circle of radius a is

the set of all points P (x , y ) whose
distance from some center C (h , k )
equals a.From the distance formula,
P lies on the circle if and only if

q
(x − h )2 + ( y − k )2 = a .

The equation of a circle with center (h, k) and radius a

(x − h )2 + (y − k )2 = a 2 .

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Distance and Circles in the Plane

Example
Find the equation for the circle of radius 2 with centered at (3,4).

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Distance and Circles in the Plane
Example
Find the center and radius of the circle

x 2 + y 2 + 4x − 6y − 3 = 0.

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Distance and Circles in the Plane

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Functions

Definition
A function f from a set D to a set Y is a rule that assigns a unique(single)
element y ∈ Y for each element x ∈ D. The relation is denoted by

y = f (x ).

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Functions

Definition
A function f from a set D to a set Y is a rule that assigns a unique(single)
element y ∈ Y for each element x ∈ D. The relation is denoted by

y = f (x ).

The element x is the input of the function and y is the value of the
function f at x.

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Functions

Definition
A function f from a set D to a set Y is a rule that assigns a unique(single)
element y ∈ Y for each element x ∈ D. The relation is denoted by

y = f (x ).

The element x is the input of the function and y is the value of the
function f at x.
The set D of all possible input values is called the domain of the
function.
The set of all values of f (x ) is called the range of the function.

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Functions
Example
Determine each of the following are functions.
a) y = x 2 + 1
b) y 2 = x + 1

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Functions
Example
Find the domain and the range of the following functions.
a) f (x ) = 5x − 3
√
b) f (x ) = 4−x
1
c) f (x ) =
1 − x2

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Graph of a function
Definition
Given a function f : X → Y , the graph of f is the set

Gf = (x , f (x )) | x ∈ X .

.

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Graph of a function
Definition
Given a function f : X → Y , the graph of f is the set

Gf = (x , f (x )) | x ∈ X .

.

Example: Graph the piece-wise defined function





 −x if x < 0

 2
f (x ) = 
 x if 0 ≤ x ≤ 1

if x > 1

1


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Graph of a function

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Vertical Line Test

The vertical line test is a visual way to determine a curve is a grap of a

function or not.

Vertical Line Test: No vertical line can intersect a graph of a function

.more than once.

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Vertical Line Test

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Vertical Line Test

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One-to-one Functions
Definition
A function f is one-to-one if every element of the range of f exactly one
element of the domain of f.

f (x1 ) = f (x2 ) implies x1 = x2 ⇔ f is one-to-one

x1 , x2 implies f (x1 ) , f (x2 ) ⇔ f is one-to-one

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One-to-one Functions
Definition
A function f is one-to-one if every element of the range of f exactly one
element of the domain of f.

f (x1 ) = f (x2 ) implies x1 = x2 ⇔ f is one-to-one

x1 , x2 implies f (x1 ) , f (x2 ) ⇔ f is one-to-one

Example: 1) f (x ) = 3x + 5

Example: 2) f (x ) = |x |

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Horizontal Line Test

The horizontal line test is a test used to determine whether a function is

one-to-one or not.

Horizontal Line Test: No horizontal line can intersect a graph of a 1-1

.function more than once.

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Horizontal Line Test

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Horizontal Line Test

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Sum, Difference, Products, Quotients and Composite
Function

▷ (f ± g )(x ) = f (x ) ± g (x )
▷ (f .g )(x ) = f (x ).g (x )
!
. f f (x )
▷ (x ) = for x ∈ Df ∩ Dg and g (x ) , 0.
g g (x )

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Sum, Difference, Products, Quotients and Composite
Function

▷ (f ± g )(x ) = f (x ) ± g (x )
▷ (f .g )(x ) = f (x ).g (x )
!
. f f (x )
▷ (x ) = for x ∈ Df ∩ Dg and g (x ) , 0.
g g (x )

Definition
If f and g two functions, the composite function f ◦ g is defined by

(f ◦ g )(x ) = f (g (x )).

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Sum, Difference, Products, Quotients and Composite
Function

▷ (f ± g )(x ) = f (x ) ± g (x )
▷ (f .g )(x ) = f (x ).g (x )
!
. f f (x )
▷ (x ) = for x ∈ Df ∩ Dg and g (x ) , 0.
g g (x )

Definition
If f and g two functions, the composite function f ◦ g is defined by

(f ◦ g )(x ) = f (g (x )).

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Sum, Difference, Products, Quotients and Composite
Function
Example
√
If f (x ) = x and g (x ) = x + 1, find f ◦ g, g ◦ f , f ◦ f and g ◦ g.

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Inverse Function

Definition
Suppose that f is one-to-one function on a domain D with range R. The
inverse function f −1 defined by

f −1 (y ) = x if f (x ) = y .

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Inverse Function

Definition
Suppose that f is one-to-one function on a domain D with range R. The
inverse function f −1 defined by

f −1 (y ) = x if f (x ) = y .

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