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)

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.)

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

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!

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

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)

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

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

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

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.

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

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

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

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

Domain: Graphical [2,∞) (-∞,∞)

Domain: Graphical (-∞,∞) [-3,∞)

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

Are They Functions? Yes No Yes No