Spring2012
KirylTsishchanka
Syllabus
LearningObjectives
We will begin by understanding what a function is and how it can be
represented geometrically as a graph. We will then learn some standard
function manipulations (algebraic combinations, compositio
Mathematical Functions - Fall 2011
Midterm Exam, October 13, 2011 In all non-multiple choice problems you are required to show all your work and provide the necessary explanations everywhere to get full credit. In all multiple choice problems you don't ha
Mathematical Functions
Sample Midterm Exam In the following problems you are required to show all your work and provide the necessary explanations everywhere to get full credit. I. (20 points) Find all real solutions of the following equations: (a) 2x2 -
Mathematical Functions
Sample Midterm Exam In the following problems you are required to show all your work and provide the necessary explanations everywhere to get full credit. 1. Find all real solutions of the following equations: (a) x2 - 3x - 8 = 0
(b
Mathematical Functions
Sample Midterm Exam In the following problems you are required to show all your work and provide the necessary explanations everywhere to get full credit. 1. Find an equation of the line with the given properties: Perpendicular to t
Section 2.1 What Is a Function?
DEFINITION: A function f is a rule that assigns to each element x in a set A exactly one
element, called f (x), in a set B . The set A is called the domain of f. The range of f is the
set of all possible values of f (x) as
Section 2.2 Graphs of Functions
DEFINITION: A function f is a rule that assigns to each element x in a set A exactly one
element, called f (x), in a set B . Its graph is the set of ordered pairs
cfw_(x, f (x) | x A
EXAMPLE:
Sketch the graphs of the follow
Section 2.3 Getting Information from the Graph of a
Function
Increasing and Decreasing Functions
DEFINITION: A function f is called increasing on an interval I if
f (x1 ) < f (x2 ) whenever x1 < x2 in I
It is called decreasing on an I if
f (x1 ) > f (x2 )
Section 2.4 Average Rate of Change of a Function
Suppose you take a car trip and record the distance that you travel every few minutes. The
distance s you have traveled is a function of the time t:
s(t) = total distance traveled at time t
We graph the fun
Section 2.5 Transformations of Functions
Vertical Shifting
EXAMPLE: Use the graph of f (x) = x2 to sketch the graph of each function.
(a) g (x) = x2 + 3
(b) h(x) = x2 2
EXAMPLE: Use the graph of f (x) = x3 9x to sketch the graph of each function.
(a) g (x
Section 1.10 Lines
The Slope of a Line
EXAMPLE: Find the slope of the line that passes through the points P (2, 1) and Q(8, 5).
Solution: We have
m=
y2 y1
51
4
2
=
=
x2 x1
82
6
3
1
EXAMPLE: Find the slope of the line that passes through the points P (2, 1
Section 1.7 Inequalities
Linear Inequalities
An inequality is linear if each term is constant or a multiple of the variable.
EXAMPLE: Solve the inequality 3x < 9x + 4 and sketch the solution set.
Solution: We have
3x < 9x + 4
3x 9x < 9x + 4 9x
6 x < 4
()
Section 1.6 Modeling with Equations
EXAMPLE: A car rental company charges $30 a day and 15c a mile for renting a car. Helen
rents a car for two days and her bill comes to $108. How many miles did she drive?
Solution: We are asked to nd the number of miles
Section 1.4 Rational Expressions
The Domain of an Algebraic Expression
The domain of an algebraic expression is the set of real numbers that the variable is permitted
to have.
EXAMPLES:
1. The domain of
2x + 4
is all real numbers except 3. We can write th
Section 1.2 Exponents and Radicals
Integer Exponents
A product of identical numbers is usually written in exponential notation. For example, 5 5 5
is written as 53 . In general, we have the following denition.
EXAMPLES:
( )5 ( ) ( ) ( ) ( ) ( )
1
1
1
1
1
Section 1.1 Real Numbers
Types of Real Numbers
1. Natural numbers (N):
1, 2, 3, 4, 5, . . .
2. Integer numbers (Z):
0, 1, 2, 3, 4, 5, . . .
REMARK: Any natural number is an integer number, but not any integer number is a natural
number.
3. Rational number