Chapter 0
Functions
§ 0.1
Functions and Their Graphs
Rational & Irrational Numbers
Definition
Rational Number: A number
that may be written as a finite
or infinite repeating decimal,
in other words, a number that
can be written in the form
m/n such that m, n are
integers
Irrational Number: A
number that has an infinite
decimal representation whose
digits form no repeating
pattern
Example

2
 0.285714 
7
3  1.73205
The Number Line
The Number Line
A geometric representation of the real numbers is shown
below.

-6
-5
-4
-3
-2
-1
2
7
0
3
1
2
3
4
5
6
Open & Closed Intervals
Definition
Open Interval: The set of
numbers that lie between
two given endpoints, not
including the endpoints
themselves
Closed Interval: The set of
numbers that lie between
two given endpoints,
including the endpoints
themselves
Example
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
1
2
3
4
5
6
4, 
x4
-6
-5
-4
-3
-2
-1
0
[-1, 4]
1  x  4
Functions
EXAMPLE
If f x   x 2  4 x  3 , find f (a - 2).
Domain
Definition
Domain of a Function: The
set of acceptable values for
the variable x.
Example
The domain of the function
f x  
is
1
3 x
3 x  0
3 x
Graphs of Functions
Definition
Graph of a Function: The set
of all points (x, f (x)) where x is
the domain of f (x). Generally,
this forms a curve in the xyplane.
Example
The Vertical Line Test
Definition
Example
Vertical Line Test: A curve in
the xy-plane is the graph of a
function if and only if each
vertical line cuts or touches the
curve at no more than one
point.
Although the red line intersects
the graph no more than once
(not at all in this case), there
does exist a line (the yellow
line) that intersects the graph
more than once. Therefore,
this is not the graph of a
function.
Graphing Calculators
Graphing Using a Graphing Calculator
Step
1) Enter the expression
for the function.
2) Enter the specifications
for the viewing window.
3) Display the graph.
Display
Graphs of Equations
EXAMPLE


1
2
Is the point (3, 12) on the graph of the function f x    x   x  2  ?
§ 0.2
Some Important Functions
Linear Equations
Equation
y = mx + b
(This is a linear function)
x=a
(This is not the graph of a
function)
Example
Linear Equations
CONTINUED
Equation
y=b
Example
Piece-Wise Functions
EXAMPLE
1  x for x  3
.
2
for
x

3

Sketch the graph of the following function f x   
Quadratic Functions
Definition
Quadratic Function:
A function of the
form
f x   ax 2  bx  c
where a, b, and c are
constants and a  0.
Example
Polynomial Functions
Definition
Example
Polynomial Function: A
function of the form
f x   an x n  an 1 x n 1    a0
where n is a nonnegative
integer and a0, a1, ...an are
given numbers.
f x   17 x3  x 2  5
Rational Functions
Definition
Rational Function: A
function expressed as the
quotient of two
polynomials.
Example
3x  x 4
g x   2
5x  x  1
Power Functions
Definition
Example
Power Function: A
function of the form
f x   x 5.2
f x   x r .
Absolute Value Function
Definition
Example
Absolute Value Function:
The function defined for
all numbers x by
f x   x
f x   x ,
such that |x| is understood
to be x if x is positive and
–x if x is negative
f  1 2   1 2  1 2
§ 0.4
Zeros of Functions – The Quadratic Formula
and Factoring
Zeros of Functions
Definition
Example
Zero of a Function: For
a function f (x), all values
of x such that f (x) = 0.
f x   x 2  1
0  x2 1
x  1
Quadratic Formula
Definition
Quadratic Formula: A
formula for solving any
quadratic equation of the
form ax 2  bx  c  0.
The solution is:
 b  b  4ac
x
.
2a
2
There is no solution if
b 2  4ac  0.
Example
x 2  3x  2  0
a  1; b  3; c  2
x
x
 3 
32  41 2
21
 3  17
2
These are the
solutions/zeros of the
quadratic function
f x   x 2  3x  2.
Graphs of Intersecting Functions
EXAMPLE
Find the points of intersection of the pair of curves.
y  x 2  10 x  9;
y  x 9
Factoring
EXAMPLE
Factor the following quadratic polynomial.
6 x  2 x3
Factoring
EXAMPLE
Solve the equation for x.
1
5 6

x x2
§ 0.5
Exponents and Power Functions
Exponents
Definition
Example
bn = b*b*b…*b
53  5  5  5
1
n
b  b
n
1
3
5 3 5
Exponents
Definition
m
n
b  b 
b

m
n

m
n
1
b
m
n

 b
1
n
b
Example
m
3
4
5  5 
m
n

1
 b
n
m
5

3
4

3
4
1
5
3
4

 5
1
4
3
5
4

3
1
 5
4
3
Exponents
Definition
b b  b
r
b
s
r
1
 r
b
rs
Example
1
3
2
3
6 6  6
4

1
2

1 2

3 3
1
4
1
2
3
3
 6  61  6

1
1

4 2
Exponents
Definition
Example
7
br
r s

b
bs
b 
r s
 b rs
7

9


4
5
4
3
1
3
5
8
7
4 1

3 3
3
3
 7  71  7
45
4
1


  95 8  98  92  9  3


Exponents
Definition
abr  a r  b r
r
ar
a
   r
b b
Example
125  271/ 3  1251/ 3  271/ 3
 3 125  3 27  5  3  15
4
10 4  10 
4




2
 16


4
5
5
Applications of Exponents
EXAMPLE
Use the laws of exponents to simplify the algebraic expression.
 27 x 
5 2/3
3
x
Compound Interest - Annual
Definition
Example
Compound Interest Formula:
A  P1  i 
n
If $700 is invested, compounded
annually at 8% for 8 years, this will
grow to:
A = the compound amount
8
(how much money you end
A  7001  0.08
up with)
8
A  7001.08
P = the principal amount
A  7001.851
invested
A  1,295.651
i = the compound interest rate
per interest period
Therefore, the compound amount
n = the number of
would be $1,295.65.
compounding periods
Compound Interest - General
Compound Interest - General
EXAMPLE
(Quarterly Compound) Assume that a $500 investment earns interest compounded quarterly.
Express the value of the investment after one year as a polynomial in the annual rate of interest r.