2. FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008

3. 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)

4. 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.)

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

6. Determine whether the following correspondences are functions: Numbers: -39 3 24 Friday Night’s Date: Juan Casandra Boris Rebecca Nelson Helga Bernie Natasha YES! NO!

7. You Do: Are these Correspondences Functions? Numbers: -636 -24 2 Numbers: -3 2 1 4 5 6 9 8 YES! NO!

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

9. DomainRange -100 -82 -64 -46 -68 No; -6 repeats DomainRange -100 -82 -64 -46 -28 Yes; D:{-10, -8, -6, -4, -2}; R:{0, 2, 4, 6, 8}

10. Testing for Functions Algebraically WWhich of these is a function? A. x 2 + y = 1 B. -x + y 2 = 1 Do you know why?

11. WWhich of these is a function? A. x 2 + y = 1 Solve for y: y = -x 2 + 1 No matter what I substitute for x, I will only get one y-value

12. WWhich of these is a function? B. -x + y 2 = 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.

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

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

15. Interval Notation: Bounded Intervals  NotationInterval Type InequalityGraph  [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

16. Interval Notation: Unbounded Intervals  NotationInterval Type InequalityGraph  (- ,b]Unbounded leftx  b ] Closed b  (- ,b)Unbounded left x < b ) Open b  [a,  )Unbounded right a  x [ Closed a  (a,  ) Unbounded right a < x ( Open a

17. Domain: Graphical [2,∞) (-∞,∞)

18. Domain: Graphical (-∞,∞) [-3,∞)

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

20. Are They Functions? Yes No Yes