1. FUNCTIONS & GRAPHS 2.1
JMerrill, 2006
Revised 2008

2. Definitions What is domain?
Domain: the set of input values (x-coordinates)
What is range?
Range: the set of output values (y-coordinates)
Relation: a pair of quantities that are related in some way (a set of ordered pairs)

3. Definitions Continued
What is a function?
A function is a dependent relationship between a first set (domain) and a second set (range), such that each member of the domain corresponds to exactly one member of the range. (i.e. NO x-values are repeated.)

4. Variable Reminders The independent/dependent variable is the x-value
The independent/dependent variable is the y-value
The independent variable is the horizontal/vertical axis on an x-y plane
The dependent variable is the horizontal/vertical axis on an x-y plane

5. Determine whether the following correspondences are functions:
Numbers:
-3 9 3
2 4
Friday Night’s Date:
Juan Casandra
Boris Rebecca
Nelson Helga
Bernie Natasha
YES!
NO!

6. You Do: Are these Correspondences Functions?
Numbers:
-2 4 2
Numbers:
YES!
NO!

7. Determine whether the relation is a function
Determine whether the relation is a function. If yes, identify the domain and range
{(2,10), (3,15), (4,20)}
Yes
Domain: {2, 3, 4}. Range: {10, 15, 20}
{(-7,3), (-2,1), (-2,4), (0,7)}
No (the x-value of -2 repeats)

8. Determine whether the relation is a function
Determine whether the relation is a function. If yes, identify the domain and range
Domain
Range
-10 -8 2 -6 4 -4 6 8
Domain
Range
-10 -8 2 -6 4 -4 6 -2 8
Yes; D:{-10, -8, -6, -4, -2};
R:{0, 2, 4, 6, 8}
No; -6 repeats

9. Testing for Functions Algebraically
Which of these is a function?
A. x2 + y = 1
B. -x + y2 = 1
Do you know why?

10. Testing for Functions Algebraically
Which of these is a function?
A. x2 + y = 1
Solve for y: y = -x2 + 1
No matter what I substitute for x, I will only get one y-value

11. Testing for Functions Algebraically
Which of these is a function?
B. -x + y2 = 1
Solve for y:
If x = 3 for example, y = 2 or -2. So each x pairs with 2-different y’s. The x’s repeat—not a function.

12. Function Notation f(x) = y
So f(x) = 3x + 2 means the same thing as y = 3x + 2
f is just the name of the function

13. Evaluating a Function Let g(x) = -x2 + 4x + 1 A. Find g(2)
B. Find g(t)
C. Find g(x+2)
A. g(2) = 5
B. g(t) = -t2 + 4t + 1
C. g(x+2) = -x2 + 5

14. Interval Notation: Bounded Intervals
Notation Interval Type Inequality Graph
[a,b] Closed a  x  b [ ]
a b
(a,b) Open a < x < b ( )
a b
[a,b) Half-open a  x < b [ )
Closed-left; a b
Open right
(a,b] Half-open a < x  b ( ]
Open-left a b
Closed-right

15. Interval Notation: Unbounded Intervals
Notation Interval Type Inequality Graph
(-,b] Unbounded left x  b ]
Closed b
(-,b) Unbounded left x < b )
Open b
[a,) Unbounded right a  x [
Closed a
(a,) Unbounded right a < x (
Open a

16. Domain: Graphical
[2,∞)
(-∞,∞)

17. Domain: Graphical
(-∞,∞)
[-3,∞)

18. Graphs: Are These Functions?
How Can You Tell?
Yes
Yes
The Vertical Line Test No No

19. Are They Functions?
Yes No
Yes No