2015 The Institute for the Professional Development of Adult Educators What Is a Function?

2015 The Institute for the Professional Development of Adult Educators What is a function? 2 Functions

2015 The Institute for the Professional Development of Adult Educators A function is a relation in which each element of the domain is paired with exactly one element of the range. Another way of saying it is that there is one and only one output (y) with each input (x). f(x)f(x) x y Functions 3

2015 The Institute for the Professional Development of Adult Educators Function Notation Output Input Name of Function Functions 4

2015 The Institute for the Professional Development of Adult Educators Functions 5 In order for a relationship to be a function... INPUT (DOMAIN) OUTPUT (RANGE) FUNCTION MACHINE EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT

2015 The Institute for the Professional Development of Adult Educators Which of the following relations are functions? 6 R= {(9,10), (-5, -2), (2, -1), (3, -9)} S= {(6, a), (8, f), (6, b), (-2, p)} T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)} No two ordered pairs can have the same first coordinate (and different second coordinates). R is a function. Is It a Function?

2015 The Institute for the Professional Development of Adult Educators 7 Input Output Domain (input) = {-3, 1,3,4} Range (output) = {3,1,-2} Function? Yes: each input is mapped onto exactly one output Is It a Function?

2015 The Institute for the Professional Development of Adult Educators 8 1. {(2,5), (3,8), (4,6), (7, 20)} 2. {(1,4), (1,5), (2,3), (9, 28)} 3. {(1,0), (4,0), (9,0), (21, 0)} Is It a Function?

2015 The Institute for the Professional Development of Adult Educators Vertical Line Test: a relation is a function if a vertical line drawn through its graph, passes through only one point. Is It a Function? 9 AKA: “The Pencil Test” Take a pencil and move it from left to right (–x to x); if it crosses more than one point, it is not a function.

2015 The Institute for the Professional Development of Adult Educators Vertical Line Test Would this graph be a function? NO Is It a Function? 10

2015 The Institute for the Professional Development of Adult Educators This is a function because no vertical line can be drawn so that it intersects the graph more than once. A function is a relation in which each element of the domain is paired with exactly one element of the range. When x = 0, y = 1 and y = 6. So, this relation is not a function. Which are functions and why? This is a function because no vertical line can be drawn so that it intersects the graph more than once. Is It a Function? 11

2015 The Institute for the Professional Development of Adult Educators Quick Review Which relations below are functions? a. Names and social security numbers b. Addresses and names c. {(2,4) (-2, 5) (3, 7)} d. {(4,1) (4, 3) (5, 6)} e. { (2,5) (3, 5) (4,5)} 12

2015 The Institute for the Professional Development of Adult Educators 13 Real World Examples of Functions Distance (output) and time (input) during a trip –For example, if you are traveling, the distance that you travel depends on the time that you spend on the road. Amount of money earned (output) and hours worked (input) –For example, if you work in a fast food outlet, your pay packet depends on the number of hours you work. Or, the amount of concrete you need to order when constructing a building will depend on the height of the building. Connection

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