Lesson 4 Representing and Understanding Functions
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1 Lesson 4 Representing and Understanding Functions Key Learning Goals I can represent relations using words, tables of values, mapping diagrams, graphs and equations I can determine if a relation is a function from its table of values, mapping diagram, graph or equation I can determine the domain and range of a function Minds On The goal of this lesson is to identify a relationship as a function. We will complete the following Investigation to examine a relationship between two variables and then determine if it is a function or not. Let's discuss: Investigate the Math, p. 4 5, Parts A K
2 A C. D. Line of Best Fit the line that best goes through all of the points on the graph To find the equation of the line we need the slope and the y intercept.
3 E. Domain the set of values for the independent variable (x) D = {7.5, 8, 8.5, 9, 9.5, 10, 10.5, 11, 11.5, 12} Note: The domain cannot be D = {7.5 12} because shoe size is not a continuous variable. ie. 7.8 is not a valid shoe size. Continuous = any real number (fraction, decimal, whole, integer) Range the set of values for the dependent variable (y) R = { } Height is continuous it can be any fraction of a centimeter. F. The relation is not a function because people can have the same shoe size but be different heights. For example there were six different people of different heights who had a size 10 shoe size. (10,158), (10, 173), (10, 170), (10, 171), (10, 170), and (10, 168) This means that there is not one distinct y value for each value of x. We can see this on the graph where points are 'stacked' vertically.
4 G&H. The line of best fit takes an average of the data and creates a rule to predict one height for a given shoe size. Therefore, the equation of the line of best fit is a function. This equation will yield only one height for a given shoe size. Conclusions 1. A relation is a set of ordered pairs that compares an independent variable (x) with a dependent variable (y). Not all relations are functions. 2. A function is a relation where each independent value (x) is paired with only one distinct dependent value (y). All functions are relations. Function: x can only be paired with one y value. Action Functions can be represented by tables of values, mapping diagrams, graphs and equations. Examples Representing Functions Table of Values Graph Mapping Diagram Equation
5 Function or Not? How to Identify a Relation as a Function Mapping Diagrams Using a mapping diagram is perhaps the easiest way to identify a relation as a function. If two arrows leave one x value, you know it is not a function. Tables of Values If you see a given value of x listed twice and there is a different value of y given for each x, the relation is not a function. Graphs To determine if a relation is a function from its graph we use the vertical line test. If the graph of the relation crosses a vertical line in two or more places, it is not a function.
6 Function or Not? Try it! Table of Values Graph Mapping Diagram Equation The Domain and Range of a Function The domain of a function is the set of all possible values of x that exist for the function f(x). The range of a function is the set of all possible values of y that exist for the function f(x). What values of x are possible for the function? What values of y are possible for the function?
7 Practice Describe the domain and range for each graph. Writing the Domain and Range Using Set Notation First you need to determine what set of numbers the values belong to. ie. Is the function defined for only whole numbers? Integers?
8 We use symbols to summarize statements. means "x is a real number." means "x is a real number, such that..." < > means "less than" means "more than" means "less than or equal to" means "more than or equal to" means "not equal to" Examples How to write the domain/range in set notation. Write each domain described below in proper set notation. a) The domain is between 4 and 5 inclusive. b) The domain is above 4 and below 20. c) The domain can be any value except 10. d) The domain is equal to or greater than 5 and can never equal 2.
9 Practice Consider the relation below. State the domain and range in: a) words b) set notation Consider the relation below. State the domain and range in: a) words b) set notation
10 Consider the relation below. State the domain and range in: a) words b) set notation Consolidate
11 Consolidate Group Practice Each group is given a set of relations. For each relation determine whether it is a function or not. State the domain and range for the functions represented graphically. Answers
12
13 Homework: 1. Review/define the following terms in your notes: relation, function, domain and range (page 5). 2. Read Examples 1 3, pages Section 1.1., p , # 1 4, 12 Section 1.4, p , #1 3, 5 Domain and Range Worksheet
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