Functions as Mappings from One Set to Another

Transcription

1 ACTIVITY. Functions as Mappings from One Set to Another As ou learned previousl, ordered pairs consist of an -coordinate and a -coordinate. You also learned that a series of ordered pairs on a coordinate plane can represent a pattern. You can also use a mapping to show ordered pairs. A mapping represents two sets of objects or items. Arrows connect the items to represent a relationship between them. When ou write the ordered pairs for a mapping, ou are writing a set of ordered pairs. A set is a collection of numbers, geometric figures, letters, or other objects that have some characteristic in common. Use braces, { }, to denote a set.. Write the set of ordered pairs that represent a relationship in each mapping. a. b. c. d Create a mapping from the set of ordered pairs. a. {(, 8), (, 9), (, 8), (8, )} b. {(, ), (9, 8), (, ), (, )} LESSON : One or More Xs to One Y M-

2 . Write the set of ordered pairs to represent each table. a. Input Output b. The mappings and ordered pairs shown in Questions through form relations. A relation is an set of ordered pairs or the mapping between a set of inputs and a set of outputs. The first coordinate of an ordered pair in a relation is the input, and the second coordinate is the output. A function maps each input to one and onl one output. In other words, a function has no input with more than one output. The domain of a function is the set of all inputs of the function. The range of a function is the set of all outputs of the function. Notice the use of set notation when writing the domain and range. WORKED EXAMPLE In each mapping shown, the domain is {,,, }. The range is {,,, }. The range is {,, }. Each mapping represents a function because no input, or domain value, is mapped to more than one output, or range value. M-8 TOPIC : Introduction to Functions

3 WORKED EXAMPLE In the mapping shown, the domain is {,,,, } and the range is {,,, }. NOTES This mapping does not represent a function.. State wh the relation in the worked eample shown is not a function.. State the domain and range for each relation in Questions and. Then, determine which relations represent functions. If the relation is not a function, eplain wh not. LESSON : One or More Xs to One Y M-9

4 Think about the mappings as ordered pairs.. Review and analze Emil s work. Eplain wh Emil's mapping is not an eample of a function. Emil M mapping represents a function. 9. Determine if each sequence represents a function. Eplain wh or wh not. If it is a function, identif its domain and range. Create a mapping to verif our answer. a.,,, 8,, Remember that a sequence has a term number and a term value. b.,,,,, c.,,,,, M- TOPIC : Introduction to Functions

5 ACTIVITY. Functions as Mapping Inputs to Outputs You have determined if sets of ordered pairs represent functions. In this activit ou will eamine different situations and determine whether the represent functional relationships. Read each contet and decide whether it fits the definition of a function. Eplain our reasoning.. Input: Sue writes a thank-ou note to her best friend. Output: Her best friend receives the thank-ou note in the mail.. Input: A football game is being telecast. Output: It appears on televisions in millions of homes.. Input: There are four puppies in a litter. Output: One pupp was adopted b the Smiths, another b the Jacksons, and the remaining two b the Fullers.. Input: The basketball team has numbered uniforms. Output: Each plaer wears a uniform with her assigned number.. Input: Beverl Hills, California, has the zip code 9. Output: There are, people living in Beverl Hills.. Input: A sneak preview of a new movie is being shown in a local theater. Output: people are in the audience. LESSON : One or More Xs to One Y M-

6 . Input: Tara works at a fast food restaurant on weekdas and a card store on weekends. Output: Tara s job on an one da. 8. Input: Janelle sends a tet message to everone in her contact list on her cell phone. Output: There are friends and famil on Janelle s contact list. ACTIVITY. Determining Whether a Relation Is a Function Analze the relations in each pair. Determine which relations are functions and which are not functions. Eplain how ou know.. Mapping A Mapping B M- TOPIC : Introduction to Functions

7 . Table A Table B Input Output. Sequence A Sequence B,,,, 9,,,,,,. Set A Set B {(, ), (, ), (, ), (, ), (, )} {(, ), (, ), (, ), (, ), (, )}. Scenario A Input: The morning announcements are read over the school intercom sstem during homeroom period. Output: All students report to homeroom at the start of the school da to listen to the announcements. Scenario B Input: Each student goes through the cafeteria line. Output: Each student selects a lunch option from the menu. LESSON : One or More Xs to One Y M-

8 ACTIVITY. Functions as Graphs A relation can be represented as a graph. A scatter plot is a graph of a collection of ordered pairs that allows an eploration of the relationship between the points.. Determine if each scatter plot represents a function. Eplain our reasoning. a. b. Output Output Input Input The vertical line test is a visual method used to determine whether a relation represented as a graph is a function. To appl the vertical line test, consider all of the vertical lines that could be drawn on the graph of a relation. If an of the vertical lines intersect the graph of the relation at more than one point, then the relation is not a function. WORKED EXAMPLE Consider the scatter plot shown. 9 8 In this scatter plot, the relation is not a function. The input value can be mapped to two different outputs, and. Those two outputs are shown as intersections to the vertical line drawn at. 8 9 M- TOPIC : Introduction to Functions

9 . Use the definition of function to eplain wh the vertical line test works. NOTES. Use the vertical line test to determine if each graph represents a function. Eplain our reasoning. a. b.. Use the cards that ou sorted in the previous lesson. Sort the graphs into two groups: functions and non-functions. Use the letter of each graph to record our findings. Functions Non-functions LESSON : One or More Xs to One Y M-

10 NOTES ACTIVITY. Functions as Equations So far, ou have determined whether a mapping, contet, or a graph represents a function. You can also determine whether an equation is a function. WORKED EXAMPLE The given equation can be used to convert ards to feet. Let represent the number of ards, and let represent the number of feet. To test whether this equation is a function, first, substitute values for into the equation, and then determine if an -value can be mapped to more than one -value. If each -value has eactl one -value, then it is a function. Otherwise, it is not a function. 9 8 In this case, ever -value can be mapped to onl one -value. Each -value is multiplied b. Some eamples of ordered pairs are (, ), (, ), and (, ). Therefore, this equation is a function. It is not possible to test ever possible input value in order to determine whether or not the equation represents a function. You can graph an equation to see the pattern and use the vertical line test to determine if it represents a function. M- TOPIC : Introduction to Functions

11 . Determine whether each equation is a function. List three ordered pairs that are solutions to each. Eplain our reasoning. a. b. If ou do not recognize the graph of the equation, use a graphing calculator to see the pattern. c. d. e. f.. Eplain what is wrong with Talor's reasoning. Talor The equation = represents a function. If two different inputs go to the same output, it can still be a function. 9 LESSON : One or More Xs to One Y M-

12 NOTES TALK the TALK Function Organizer. Complete the graphic organizer for the concept of function. Write a definition for function in our own words. Then, create a problem situation that can be represented using a function. Finall, create a table of ordered pairs and sketch a graph to represent the function. Definition Problem Situation Function Graph Table/ Ordered Pairs M-8 TOPIC : Introduction to Functions