Review Homework
Pages 123-124
7) D={1,3,5,7} R={2,4,6,8} function both
8) D={-3,0,2,4,5} R={0,2,4,5} function onto
9) D={-4,-2,1,3} R={-4,1,3,5} not a function
10) D={-1,1,3,5,7} R={-4,-2,0,2,4}function both
11) -10
12) 11
13) 2
14) -3y+2
15) 3a+2
16) -6w+2
Properties of Exponents Review
Property
Product of Powers
Definition
x x  x
a
b
a b
Quotient of Powers
xa
a b
x

0
,

x
If
xb
Negative Exponent
x a 
1
1
and
 xa , x  0
a
a
x
x
Power of a Power
x 
Power of a Product
xy
Power of a Quotient
x
xa
   a , y  0, and
y
 y
a b
x
a
z4×z3=z4+3
76 ×7 2= 76+2
8
65
5 3
2 x
8 4
4

6

6
,

x

x
63
x4
3 6 
4 
2 3
ab
x y
a
a
1
1
9
,

b
36 b 9
 
3
 4 23  46 , x12  x123  x 36
2 y 3  23 y 3  8 y 3 ,  pq 2  p 2 q 2
a
a
a
x
 y y
     or a , x  0, y  0
x x
 y
Zero Power
Examples
a
x 0  1, x  0
5
y5  b 
 y
   5 , 
x a
x
7
80  1
1928492034756 0  1
a7
 7
b
2.6 Special Functions
• Piecewise function
• Step Function (Greatest Integer Function)
• Absolute Value Function
• Whenever a linear function has the form
y = ax + b (y=mx + b) and b = 0 and
a ≠ 0, it is called a direct variation.
[y=ax]
• A constant function is a linear function
in the form y = ax + b where a = 0.
[y=b]
• An identity function is a linear function
in the form y = ax + b where a = 1 and
b = 0. [y=x]
The Piecewise Function
A function that is comprised of two or
more expressions. A solid point
indicates the point is included in the
graph while a hollow or open point
shows the point is not included in the
graph.
Graphing Piecewise Functions
x  4

g  x   2x  5
 x  3

x  4
4  x  1
x 1
Domain -  ,  
Range -  , 7 ]
3
1
 x2
gx  2
 x  4

1
7  x  4
4  x  0
0x5
5x7
Domain - (-7, 7]
Range - (-4, -2), [-1, 4]
 1
 3 x

h  x   x  1
x  4

 x  3
6  x  3
3  x  0
0x3
3x7
Domain - [-6, 7]
Range - [-4, 2], (4, 7)
How to write an equation from a graph
http://www.youtube.com/watch?v=3TrrGoa1-w&feature=player_embedded
Domain -  ,  
Range -  , 4
Domain - [-1, 5]
Range - [-5, 3]
Not a Function! Why?
Step Functions - consists of line segments
 1,if 0  x  1
 2,if 1  x  2
f ( x)  
3,if
2

x

3

 4,if 3  x  4
 1,if 0  x  1
 2,if1  x  2
f ( x)  
 3,if 2  x  3
 4,if 3  x  4
Graph :
 1,if  4  x  3

 2,if  3  x  2
f ( x)  
3,if

2

x


1


 4,if  1  x  0
• Step functions are functions depicted in
graphs with open circles which mean that
the particular point is not included.
– Example:
• A type of step function is the greatest integer
function which is symbolized as [x] or [[x]]
and means “the greatest integer not greater
than x.”
“Gauss”
– Examples: [8.2] = 8 [-3.9] = -4
[5.0] = 5
[7.6] = 7
Sample value
Floor
Ceiling
12/5 = 2.4
2
3
2/5 = 0.4
2.7
2
3
0.7
−2.7
−3
−2
0.3
−2
−2
−2
0
Fractional part
Sample value
Floor
Ceiling
12/5 = 2.4
2
3
2/5 = 0.4
2.7
2
3
0.7
−2.7
−3
−2
0.3
−2
−2
−2
0
Fractional part
Homework
• Page 105 # 13-23 (odd)
• Page 107 #45-48