R2: Real Numbers and their Properties
* sets of numbers
1. Natural numbers: N= cfw_1, 2,3,4,5,.
2. Whole numbers = cfw_ 0,1, 2, 3, 4, .
3. Integers: =Z
4. Rational numbers Q
Is the setof numbers that can be written in the form
where
1. a,b are integers.
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2.3 INTRODUCTION TO
FUNCTIONS
Definition of a Relation
A relation is any set of ordered pairs.
The set of all first coordinate of the ordered pairs is called the
domain ( )of the relation,
The set of all second coordinate of the ordered pairs i
5.2 Trigonometric Functions:
In aright angled triangle far an acute angle
(theta)
Hypotenuse
opposite
to
adjacent
to
Ex: Find the values of the six trigonometric functions of angle
passes through the point
1.
solution:
2.
\
solution;
Math002 Page 1
whose
Chapter 5
5.1
Angles:
An angle consists of two rays or two segments in a plane with a common end point
moves anticlockwise
moves clockwise
*The degree measure of an angle
1 rotation =360 degrees
* Types of angles:
Definition:
1. Complement angles: their s
4.3 Logarithmic Functions
Definition:
For all real numbers y and all positive numbers
a and x where
if and only if
argument
base
EX: Solve each equation
1.
2.
3.
Definition:
If
and
then
defines the logarithmic function with base a
Note:
The exponential fu
4.2 Exponential Functions:
Exponential properties
Ex: Evaluate
Ex: If
find
* Exponential Functions
1. If
,
Domain =
Range =
key
points
Math002 Page 1
1. If
,
Domain =
Range =
key
points
2. If
Domain=
Range =
key
points
Ex, Graph each function, give the do
4.5 Exponential and Logarithmic Equations
* Properties of Logarithms
If
then
if and only if
Ex: Solve:
1.
Solution:
2.
Solution:
3.
Solution:
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4.
Solution:
5.
Solution:
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6.
Solution:
let
no solution
7.
Solution:
let
Math002 Pag
4.1 Inverse Functions
Onetoone Functions:
A function
i.e. each xvalue corresponds to only one yvalue, and each yvalue correspond to
only one Xvalue
Note: A onetoone function is either increasing or decreasing
Ex: Decide whether each function is 1
4.4 Evaluating Logarithms and change of Base Theorem
* common Logarithms
for all positive numbers x
Note:
If
if
if
* Natural Logarithm
for all positive numbers x
* Change of base Theorem
for any positive real numbers x, a and b
Ex: Evaluate
1.
2.
3.
4.
5.
103 Hyperbolas
In the graph of each hyperbola considered so far, the center is the origin and the asymptotes pass
through the origin. This feature holds in general; the asymptotes of any hyperbola pass through the
center of the hyperbola. Like an ellipse
10.2 Ellipses
Definition: An ellipse is the set of points in a plane the sum of whose distances from two
fixed points called (foci) is constant
horizontal ellipse
vertical ellipse
An ellipse centered at
With vertical
major axis of length 2a has equation
v
10.1 Parabolas
Definition:
AParabola is a set of points whose distance from a fixed point called (focus) equals its
distance from a fixed line called (directrix)
FM = MD
Screen clipping taken: 7/24/2012 10:46 PM
Ex: Find the vertex, focus, axis, directrix
9.7 Properties of Matrices
a matrix (plural matrices) is a rectangular array of numbers, symbols, or
expressions, arranged in rows and columns.
The individual items in a matrix are called its elements or entries
The size of a matrix is the number of rows
9.8 Matrix Inverses

The identity matrix
Ex: Let
show that
Solution:
Multiplicative inverses:
If A is an nxn matrix and
there is a matrix A such that
is called the inverse of matrix A
Finding the inverse of a matrix A
1. Form the matrix
2. Perform the
9.5 Nonlinear systems of equations
1. .solving a system of two equations with two variables
with one of the equations is Linear
Ex: solve
solution:
take the linear equation and solve for x or for y
replace it in equation 1 which is the equation
that we di
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1.2 FORMULAS
AND
APPLICATIONS
Formula
is an equation that express known relationships between two or
more variables.
l
l
A=lw
P=2l+2w
w
A=l2
l
P=4l
A r 2 , C 2r
b
a
b
a
c
A
1
ac, P a b c
2
c
A ac, P 2b 2c
1
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Ex1. Solve the formula 2l + 2
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1.8 ABSOLUTE VALUE
EQUATIONS AND
INEQUALITIES
Absolute Value Equations
We need to rewrite Absolute Value Equations without Absolute
Value Bars.
If c is a any nonnegative real number and X represents any
algebraic expression, then
X=c
if and onl
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1.1 LINEAR EQUATIONS
Linear Equations
An equation is a statement about the equality of two expressions
Definition of a Linear Equation
A linear equation in the single variable x is an equation that can
be written in the form
ax + b = 0
where a a
3.5 Graphing Rational
Functions
1
Def.:
P( x)
A rational function is a function of the form f(x) =
,
Q( x)
where P(x) and Q(x) are polynomials and Q(x) = 0.
Example: f (x) =
1
is defined for all real numbers except x = 0.
x
x
f(x)
x
f(x)
2
0.5
2
0.5
1
1
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3.1 QUADRATIC FUNCTIONS
1
Def.:
Let a, b, and c be real numbers a 0. The function
f (x) = ax2 + bx + c
is called a quadratic function.
The graph of a quadratic function is a parabola). (
Every parabola is symmetrical about a line called the axi
3.4 POLYNOMIAL FUNCTIONS
1
All polynomial functions have graphs that are smooth continuous
curves )(
A smooth curve: no sharp corners
A continuous curve: no breaks, holes, or gaps.
continuous
smooth
polynomial
not continuous
not polynomial
continuous
n
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2.1 RECTANGULAR
COORDINATE SYSTEM
AND GRAPHS
Plotting Points in the Cartesian Coordinate System:
Plot: We move from the origin and plot points in the following way:
y
A( 3, 5)
5
A( 3, 5): 3 units left, 5 units up
4
Quadrant
B(2, 4): 2 units righ
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2.2 CIRCLES
CIRCLES, THEIR EQUATIONS, AND THEIR GRAPHS
Definition of a Circle
A circle is a set of all points in a plane that are a fixed distance
from a fixed point called the center. The fixed distance from the
circles center to any point on t
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2.7 PROPERTIES OF GRAPHS
1
I) Symmetry )(
Symmetry with Respect to the xaxis: If (x,y) on the graph,
then (x, y) is also in the graph
Symmetry with Respect to the yaxis: If (x,y) on the graph,
then (x, y) is also in the graph
Symmetry with R
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1.4 QUADRATIC EQUATIONS
Definition of a Quadratic Equation
A quadratic equation in x is an equation that can be written in the
standard form
ax2 + bx + c = 0
where a, b, and c are real numbers with a 0.
A quadratic equation in x is also called a
2.8 FUNCTION OPERATIONS
AND COMPOSITIONS
1
Def.:
Let f and g be two functions.
( f + g)(x) = f (x) + g(x)
D f g D f Dg
( f g)(x) = f (x) g(x)
D f g D f Dg
( f g)(x) = f (x) g(x)
D fg D f Dg
f
f ( x)
( x)
, g ( x) 0
g
g ( x)
D f D f Dg
g
except
x g ( x)
2.42.5
LINEAR FUNCTIONS
1
Def:
A function of the form
f ( x) mx b, m 0 where
m and b are real numbers is called a linear function
Slopes of Lines )(
P2 ( x2 , y2 )
Change in y
P ( x1 , y1 )
1
Change in x
The slope = m =
change in y y y2 y1
change in x x
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1.6 OTHER TYPES OF EQUATIONS
The Other Types of Equations Are:
1. Polynomial Equations
2. Rational Equations
3. Radical Equations
4. Equations that are Quadratic in Form
1. Polynomial Equations
Some polynomial equations that are neither linear no
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2.6 Graphs of Basic functions
The Greatest Integer Function
f (x) = int(x) or [x] greatest integer less than or equal to x.
y
If 2 x < 3, then int(x) = 2
If 1 x < 2, then int(x) = 1
If 0 x < 1, then int(x) = 0
If 1 x < 0, then int(x) = 1
If 2 x <
MATH001
PAGE 1
CODE 000
MAJOR EXAM 1 TERM 152
1) Let U = all whole numbers < 10
be the universal set.
If M = all even natural numbers < 7 , N = 1, 3, 5, 7, 9
R = 2, 3, 5, 6, 8 , then ( N
A)
R)
and
M =
M
B) N
R
C)
1, 3, 5, 7, 8, 9
D)
0, 3, 5, 6, 8, 9
E) M