1. Mrs.Volynskaya Relations Domain Range
A relation is a mapping, or pairing, of input values with output values.
The set of input values is called the domain.
The set of output values is called the range.

2. Domain is the set of all x values.
Domain & Range
Domain is the set of all x values.
Range is the set of all
y values.
{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}
Example 1:
Domain-
Range-
D: {1, 2}
R: {1, 2, 3}

3. A Relation can be represented by a set of ordered pairs of the form (x,y)
Quadrant II
X<0, y>0
Quadrant I
X>0, y>0
Origin (0,0)
Quadrant IV
X>0, y<0
Quadrant III
X<0, y<0

4. Plot:
(-3,5)
(-4,-2)
(4,3)
(3,-4)

5. Most equations have infinitely many solution points.
Every equation has solution points (points which satisfy the equation).
3x + y = 5
(0, 5), (1, 2), (2, -1), (3, -4)
Some solution points:
Most equations have infinitely many solution points.
Page 111

6. Ex 5. Graph y = x² - 5 x
x² - 5 y -3 -2 -1 1 2 3

7. Functions In order for a relationship to be a function…
EVERY INPUT MUST HAVE AN OUTPUT
TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT
ONE INPUT CAN HAVE ONLY ONE OUTPUT
INPUT
Functions
(DOMAIN)
FUNCTION MACHINE
OUTPUT
(RANGE)

8. Identify the Domain and Range. Then tell if the relation is a function.
Input Output 4
Function?
Yes: each input is mapped
onto exactly one output
Domain = {-3, 1,3,4}
Range = {3,1,-2}

9. Identify the Domain and Range. Then tell if the relation is a function.
Input Output 4
Domain = {-3, 1,4}
Range = {3,-2,1,4}
Notice the set notation!!!
Function?
No: input 1 is mapped onto
Both -2 & 1

10. The Vertical Line Test
If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function.
Page 117

11. Use the vertical line test to visually check if the relation is a function.
(4,4)
(-3,3)
(1,1)
(1,-2)
Function?
No, Two points are on
The same vertical line.

12. Use the vertical line test to visually check if the relation is a function.
(-3,3)
(1,1)
(3,1)
(4,-2)
Function?
Yes, no two points are
on the same vertical line

13. YES!
Function? #1

14. #2 Function? YES! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals
Another cool function: abs(x) + 2sin(x)

15. #3 Function? NO! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals
Another cool function: abs(x) + 2sin(x)

16. #4 Function? YES! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals
Another cool function: abs(x) + 2sin(x)

17. #5
Function?
NO!

18. YES!
Function? #6
This is a piecewise function

19. Function? #7 NO! D: all reals R: [0, 1]
Another cool function: y = sin(abs(x))
Y = sin(x) * abs(x)

20. #8 Function? NO! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals
Another cool function: abs(x) + 2sin(x)

21. YES! #9
Function?

22. Function?
#10
YES!

23. Function?
#11
NO!
D: [-3, -1) U (-1, 3]
R: {-1, 1}

24. YES!
Function?
#12
D: [-3, -1) U (-1, 3]
R: {-1, 1}

25. Function Notation
“f of x”
Input = x
Output = f(x) = y

26. (x, y) (x, f(x)) (input, output) y = 6 – 3x f(x) = 6 – 3x x y x f(x)
Before…
Now…
y = 6 – 3x
f(x) = 6 – 3x x y x
f(x) -2 -1 1 2 12 -2 -1 1 2 12
(x, y)
(x, f(x)) 9 9 6 6 3 3
(input, output)

27. g(2) = 2 3 g(5) = Find g(2) and g(5).
Example 7
Find g(2) and g(5).
g = {(1, 4),(2,3),(3,2),(4,-8),(5,2)}
g(2) = 2 3
g(5) =

28. Example 9.
f(x) = 2x2 – 3
Find f(0), f(-3), f(5a).

29. F(x) = 3x2 +1 Find f(0), f(-1), f(2a). f(0) = 1 f(-1) = 4
Example 10.
F(x) = 3x2 +1
Find f(0), f(-1), f(2a).
f(0) = 1
f(-1) = 4
f(2a) = 12a2 + 1

30. D: All real numbers except -3
What is the domain?
g(x) = -3x2 + 4x + 5
Ex.
D: all real numbers
x + 3  0
Ex.
x  -3
D: All real numbers except -3

31. What is the domain? h x ( ) = - 1 5 f x ( ) = + 1 2 x - 5  0 x + 2 0
Ex.
D: All real numbers except 5
Ex. f x ( ) = + 1 2
x + 2 0
D: All Real Numbers except -2

32. What are your questions?