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FUNCTIONS & GRAPHS 2.1

FUNCTIONS & GRAPHS 2.1. JMerrill, 2006 Revised 2008. 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).

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FUNCTIONS & GRAPHS 2.1

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  1. FUNCTIONS & GRAPHS2.1 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. Numbers: -3 9 3 2 4 Friday Night’s Date: Juan Casandra Boris Rebecca Nelson Helga Bernie Natasha Determine whether the following correspondences are functions: YES! NO!

  6. Numbers: -6 36 -2 4 2 Numbers: -3 2 1 4 5 6 9 8 You Do: Are these Correspondences Functions? YES! NO!

  7. 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. If yes, identify the domain and range 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 • NotationInterval TypeInequalityGraph • [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 • NotationInterval TypeInequalityGraph • (-,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? No Yes Yes No

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