Relations and Functions 9.2 Relations 9.5 Functions
Dec 15, 2015
Relations and Functions
9.2 Relations
9.5 Functions
F
E
Line Slope Y-intercept SI
Y=mx+b
SF
Ax + By=C
E
F
You may count the slope
or
Pick two points and use calculator
Ex 1
Find the distance between the points P(2, 2) and Q (4, -6).1, 1( )x y 2, 2( )x y
2 22 1 2 1( ) ( )d x x y y
2 2( ) ( )d 4 2 26
2 2( ) ( )d 2 8
d 4 64
68 d
Work inside parentheses
Do powers before doing the addition!!!!!!
Be careful Of Signs!!!
(neg)2 = POS.
68 d
) 2 10
) 68
) 2 10
) 2 17
A
B
C
D
8.2462116.32455532
8.246211
Ex 5
2 1midpoint2
x xx
2 1midpoint2
y yy
midpoint midpoint( , )x y
Find the midpoint whose endpoints are (1, -2) and (-17, 16)1, 1( )x y 2, 2( )x y
midpoint
2x
17 1
216
8
( , )8
midpoint
2y
16 2
214
7
7
Ex 8 M(-8, 7) is the midpoint of RS. If S has a coordinates (-6, 8),
find the coordinates of R.
2, 2( )x y
2
6 1x
16 x 16
( , )10
2
8 1y
6
,( )m mx y1, 1( )x y
R (x1, y1) S (-6, 8)M(-8, 7)
8 7(2) (2)
6 6
1x 10
(2) (2)
18 y 148 8
1y 6
Topics
Domain Range Mapping Inverse of a
Relation
Functions (relation/graphs)
Function Notation Evaluate Function
Relations (sets of ordered pairs)
Ways to display a relationList the set of ordered pairsGraphingTableMapping
X Y
2 2
5 7
-1 4
9 6
Table
Mapping
2 2
5 7
-1 4
9 6
4
2
-2
-4
-5 5
Graphing
2 2
5 7
-1 4
9 6
2 2
5 7
-1 4
9 6
2 2
5 7
-1 4
9 6
2 2
5 7
-1 4
9 6
Domain Range The domain of a
relation is the set of all the first coordinates from the ordered pairs.
The x values
The Range is the set of all the second coordinates from the ordered pairs.
The y values
A set of ordered pairs
{(2,-2), (-2,3), (5,7), (4,8)}
Domain
{ 2, -2, 5, 4}
Range
{-2, 3, 7, 8}
Ordered pairs
inparentheses
Sets denotedby brackets
Ex 1 List the ordered pairs, domain, range, and inverse
X Y
0 2
1 6
-5 -2
12 3
A set of ordered pairs{(0,2), (1,6), (-5,-2), (12,3)}
Domain{ 0, 1, -5, 12}
Range{2, 6, -2, 3}
Inverse{(2,0), (6,1), (-2, -5), (3, 12)}
What is domain?
X-values
What is Range?
y-values
Inverse---switch x and y
Ex 2 List the ordered pairs, domain, range, and inverse
X Y
8 -3
4 9
1 1
-13 7
A set of ordered pairs{(8,-3), (4,9), (1,1), (-13,7)}
Domain{ 8, 4, 1, -13}
Range{-3, 9, 1, 7}
Inverse{(-3,8), (9,4), (1,1), (7,-13)}
What is domain?
X-values
What is Range?
y-values
Inverse---switch x and y
Definition of a function
Function is a special relation where each “x” is paired with exactly one “y”
{(2,3) (4,5) (2,8)}
{(5,8), (2, 9), (3,9)}
Another way to say it X’s are girls y’s are boys Girls can’t cheat on a
boy but a boy can cheat on a girl
Not a function
Yes, a functionBoys can cheat
Examples 3-6
Which are functions?
#3 {(-9,4) (2,6) (3,8), (3,9)}
#4 {(4,6), (12, 9), (2,3), (4,6)}
#5 {(1,3) (14,15) (12,8), (14,5)}
#6 {(-5,8), (7, 8), (3,9)}
NO
YES
NO
YES
Vertical Line Test-- Can’t touch line more than once or not a function
Not a function
Yes! It’s a function
Not a function
Examples 7-10Examples 7-9
#7
______
#8
___
#8
_____YES
NO
YES
# 9
Function Notation
y=2x + 6 F(x) =2x + 6 Read “f of x” Represents the
value in the range for that particular value of x
F(3) is the function value for f for x=3.
Plug the 3 in for x F(3) = 2(3) + 6 F(3)= 6 + 6 = 12 Can write it as
(3,12)
Ex 10-12 2( ) 3g x x x
( ) 3 2f x x
10) ( 1)f
( 1) 3( 1) 2f
3 2
1
11) ( 3)g 112) ( )
3f
2( 3) 3( 3) ( 3)g 1 1( ) 3( ) 23 3
f
3(9) 3 1 2
327 3 24