Download Chapter Two: Numbers and Functions Section One: Operations with

Chapter Two: Numbers and Functions
Section One: Operations with Numbers
Every number that we have studied up to this point can be called a real number. However, later in
chapter five we will study imaginary numbers. Every real number can be classified as either rational or
irrational.
Irrational Numbers – a decimal that continues infinitely without any repetition
(ex.  , 2 , 3 , 2.131131113…)
Rational Number – all real numbers that aren’t irrational (including terminating and repeating decimals)
Since the set of rational numbers seem to include so many numbers, we can divide them even farther
into natural numbers, whole numbers, and integers.
Natural Numbers (Counting Numbers) – 1, 2, 3, 4, 5…
Whole Numbers – 0, 1, 2, 3, 4, 5…
Integers - …-3, -2, -1, 0, 1, 2, 3…
Notice that none of these include fractions or decimals.
EX1: Classify the following numbers as many ways as possible:
1. 56
2. 4.27
3. 0.12112111211112…
All real numbers possess the following properties
Closure Property – When you add or multiply real numbers you get another real number
Commutative Property – you can add or multiply in any order
(ex. 2  3  4  3  4  2
abc  bca )
Associative Property – Parentheses can be moved when adding or multiplying
(ex.  2  3  4  2   3  4  ab  c  a  bc  )
Identity Property – Adding zero or multiplying by one gives you the same number
Inverse Property – Adding a number to its opposite gives you zero. Multiplying a number by its inverse
gives you one.
Distributive Property – When something is being multiplied by multiple terms in parenthesis…
(ex. 2( x  3)  2  x  2  3 )
EX2: What property is being illustrated?
1. 1 a  a
3.
x y
 1
y x
y  2 x  2 xy
4.
5.
m( x 2  1)  mx 2  m
b  ( b )  0
2.
When simplifying algebraic expressions we must follow the four steps of PEMDAS.
1. Parenthesis
2. Exponents
3. Multiply/Divide
4. Add/Subtract
EX3: Use order of operation to simplify the expressions
1.
2.
3.
4.
6(11  32 )
8
18  9  2
2[14  3(6  1) 2 ]
14
24 
52
Section 2: Properties of Exponents
An exponent is a shorthand way of writing repeated multiplication
35  3  3  3  3  3  243
In the expression 35 , 3 is the base and 5 is the exponent or power.
Anything raised to the zero power is always one. (ex. 30  1 , x 0  1 , ( x 2  3)0  1 )
When dealing with negative powers, we want to make them positive. When a number or variable that is
raised to a negative power is moved across a fraction bar, the power becomes positive.
(ex. 32 
1 1
 ,
32 9
x 3 
1
,
x3
4 1 
1 1
 )
41 4
EX1: Evaluate the following exponents
1. 90
2. 54
3. 5 4
There are some basic exponent rules that we need to know when working with powers
1. Product of Powers – When multiplying like bases, add exponents
34  32   3  3  3  3   3  3  3  3  3  3  3  3  36
2. Quotient of Powers – When dividing like bases, subtract exponents
(Look to see if there are more on the top or more on the bottom)
45 4  4  4  4  4 4  4  4  4  4 42


  42
3
4
444
4  4  4
1
3
9
999
9  9  9
1


 4
7
9
9  9  9  9  9  9  9 9  9  9  9  9  9  9 9
3. Power of a Power – When a power is raised to another power, multiply exponents
x 
3 4
 x3  x3  x3  x3  x12
4. Power of a Product – When a product is all raised to a power, distribute the power
 3x y 
2
4 3
 3x 2 y 4  3x 2 y 4  3x 2 y 4  33   x 2    y 4   27 x8 y12
3
3
5. Power of a Quotient – When a quotient is all raised to a power, distribute the power
3
2
 3x 2  3x 2 3x 2 3x 2 3   x 
27 x8

 4   4  4  4 
3
y
y
y
y12
 y 
 y4 
3
3
Note: Number One Error!!!
Notice that we are not raising sums to powers. This requires FOIL and will be discussed in a later
chapter. (ex. ( x  3)2  x 2  32 )
EX2: Simplify the following expressions
9a 2b3  2a5b 3 
1.
  m9 n 
2.  10 6 
 5m n 
 3x 
 
 y 
3.
4
2
3
 x 8 

3
  xy  
2
(Hint: Fractions raised to a negative power can be flipped to make the power positive)
Before discussing the next topic, we need to discuss radicals or roots:
4  2 (What can we multiply twice to give us four?)
Cube or Third Root: 3 64  4 (What can we multiply three times to give us sixty-four?)
Square Root:
5
Fifth Root:
81  3 (What can we multiply four times to give us 81?)
32  2 (What can we multiply five times to give us 32?)
4
Fourth Root:
Rational (Fractional) exponents can be rewritten in the following ways:
1
1
x2  x
2
x3 
 
3
2
x
3
x5 
 
5
1
1
x3  3 x
x4  4 x
3
x
5
x9 
 
9
x5  5 x
5
x
EX3: Evaluate the following expressions
1
1.
25 2
2.
27 3
2
3
3. 814
4. 16

3
2
EX4: Use a calculator to evaluate the following decimal powers
1. 230.25  141.53
2. 852.64  243.4
7
x8 
 x
8
7
Section Three: Introduction to Functions
A book club charges $5 to join and then $10 for each book purchased. Write an equation relating x, the
number of books purchased, and y, the total cost. Make a chart giving the cost of 0, 1, 2, 3, 4, and 5
books. Do the prices ever repeat? Will they ever repeat?
Since each x has only one y that corresponds with it, this is an example of a function. A function is a
special type of relation in which each element of the domain (the x’s) is paired with only one element of
the range (the y’s).
EX1: State whether the following relations represent a function.
Domain, x Range, y
1.
1
5
2
9
3
4
Domain, x Range, y
1
5
1
9
13
3
17
3
2.
Domain, x Range, y
1
5
2
5
13
3
5
17
4
5
3.
Think about the graphs of the previous three problems. How can we tell by looking at a graph is
something is a function? If one point is directly over another, this means that an x-value is paired with
multiple y’s. This would not be a function. Therefore, if a vertical line can be drawn anywhere on the
graph and it hits more than one point, the relation would not be a function.
EX2: Do the following graphs represent functions?
a.
b.
EX3: Let S be the set of all people in Scottsboro who owns a car and V be the set of vin numbers for the
cars. If S is the domain and V is the range, does this relation represent a function?
EX4: State the domain and range of the following functions.
a.
b.
Another name for the variable x in an equation is the independent variable. Another name for y is the
dependent variable. This is because you choose your x’s independently but the y value depends on the
x-value.
If an equation represents a function, we have a special way of writing it called function notation.
y  2 x  4  f ( x)  2 x  4
y  3x  2  g ( x)  3x  2
Note: f ( x ) does not mean f times x. It is simply another way to write y and it tells us that the letter in
parenthesis is the independent variable of the equation.
The purpose of function notation is simply shorthand.
“If y  2 x  4 then find the value of y when x  3 ” can now be replaced with “If f ( x)  2 x  4 then
find f (3) .” Notice that instead of f ( x ) we now have f (3) . This simply means replace x with 3 in the
equation.
x
f ( x)  2 x  4
f ( x)
0
f (0)  2  0   4
4
1
2
10
f (1)  2 1  4
f (2)  2  2   4
f (10)  2 10   4
EX5: Evaluate each function for the given value of x.
1. g ( x)  x 2  3x  2 for x  1 and x  2
2.
r ( x)  2 x  9 for x  5 and x  4
6
8
24
Section Four: Operations with Functions
The same way that we can add, subtract, multiply, and divide numbers and expressions, we can do the
same for functions.
EX1: Let r ( x)  x 2  2 x  9 and t ( x)  6 x 2  8 , find r  t and r  t
EX2: Let a( x)  2 x 2 and b( x)  2 x  4 , find r  t and
r
(state the domain restrictions)
t
Gas costs $3.89 per gallon. Represent this as the function c ( x ) where x is the number of gallons
purchased. It takes approximately 3 gallons of gas for me to drive to and from work each day. Represent
this function as g ( x ) where x is the number of days that I drive to work. How much does it cost me to
drive to work in a five day week? For 10 days? For 180 days?
We could save a step by finding the composition of the two functions. Below is a diagram showing how a
composition works.
We can write compositions in the following way:
When we compose g ( x ) into f ( x ) :
f  g ( x)   f
When we compose f ( x ) into g ( x ) :
g
g  f ( x)   g f
EX3: Let f ( x)  x 2  4 and g ( x)  2 x , find f
g and g f
EX4: Let f ( x)  x 2 , g ( x)  3x , and h( x)  x  5 , find the following:
1. g h
2. f f
3.
4.
5.
6.
 f f  2
 f g h
 f g  h   0 
f  g h
Section Five: Inverses of Functions
The inverse of a function will undo what the original function did. For example, let’s say that you plug
the numbers 2, 3, and 4 into a function and you get 7, 8, and 9. This means that if you plugged 7, 8, and
9 into the function’s inverse you would get 2, 3, and 4. In other words, we know that plugging the
domain into a function will give you elements of the range. On the other hand, plugging elements of the
range into an inverse function would give you the corresponding domain values.
To find the inverse of a function simply switch x and y.
EX1: Find the inverse of the following relations. State whether the original relation was a function and
whether the new relation is a function.
1. 1, 2 4, 2  3, 2 
2.  2, 4 3, 4 8, 5




Note: The inverse of f ( x ) can be written as f 1 ( x)
EX2: Find the inverse of the following functions
1. f ( x)  4 x  5
2.
g ( x) 
2x  3
5
Let’s look at the plots of the following points along with their inverses:
1, 2 4, 23, 2 2, 43, 4 8, 5
Do we notice the relationship between the points and the line y  x .
A graph and its inverse will always be symmetrical across the line y  x
Also, because of this fact, we can tell if a graphs inverse will also be a function. Since a graph is not a
function if any vertical line hits more than one point, its inverse will not be a function if any horizontal
line will hit more than one point on the graph.
EX3: Does the following graphs represent functions? Would the inverse be a function?
a.
b.
There is also a way we can use compositions to determine if two functions are inverses:
If f g  x and g f  x , then the two functions are inverses of each other.
EX4: Show that f ( x)  4 x  3 and g ( x ) 
1
3
x  are inverses of each other.
4
4
Section Six: Special Functions
Mandy works as a cashier where she earns $6 for each hour that she works. However, if she earns $9 for
each hour that she works over forty hours. Since her rate depends on the amount of hours she works,
we cannot define this situation with an ordinary function. We could, however, define it with a piecewise
function. A piecewise function consists of different function rules for different domain sections. The
example could be defined in the following way:
6h, if 0  h  40
w(h)  
9h  120, if h  40
The function says that for 0 to 40 hours Mandy’s pay is based on the function w(h)  6h . However,
when she works more than 40 hours we will use the function w(h)  9h  120 .
EX1: If
a.
b.
c.
d.
3x, if 0  x  2

find the following values.
f ( x)  6, if 2<x  4
 x  10, if 4  x  6

f (0)
f (2)
f (5)
f (7)
3x, if 0  x  2

EX2: Graph f ( x)  6, if 2<x  4
 x  10, if 4  x  6

The Greatest-Integer function, f ( x)   x or f ( x)   x  , is sometimes called the rounding down
function. We simply round the value of x down to the next integer. (The Greatest-integer function is
sometimes called the floor function).
EX3: Evaluate if f ( x)   x
a.
b.
c.
f (1.3)
f (6.789)
f (4)
The opposite of the Greatest-Integer Function is the Rounding-up Function, f ( x)   x  . We simply
round the value of x up to the next integer. (The rounding-up function is sometimes called the ceiling
function.)
EX4: Evaluate if f ( x)   x 
a.
b.
f (1.3)
f (6.789)
Because of the nature of their graphs, both of these functions are called step functions.
The graph of the absolute-value function is v-shaped. The domain is all real numbers and the range is all
numbers greater than or equal to zero. (The absolute value function is entered into the calculator using
abs() ).
f ( x)  x
EX5: Describe the transformations performed on the parent function f ( x)  x and graph the new
function. Then graph the new functions inverse on the same graph.
a. f ( x)  3 x  2
b.
f ( x)  
1
x 3
2