U1D9 Have out: Bellwork: total: +3 +3 +3 +3 Assignment, pencil, red pen, highlighter, graph paper NB, Have out: Bellwork: total: 1. Sketch the subset on the real number line. a) b) +3 +3 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 2. Given the graph, write the interval using inequality notation. a) b) -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 +3 +3

You have five minutes to complete Practice #1 and #2 on the first page of today’s packet. Practice #1: Find the domain and range of each relation. Then create a mapping and state whether or not the relation is a function. a) {(–2, 2), (5, 2), (–1, 2), (0, 2), (–4, 2), (2, 2)} b) {(4, 5), (4, 2), (4, –1), (4, 3), (4, 0), (4, –2)} Practice #2: Graph the examples on the provided set of axes on the worksheet.

Practice #1: Find the domain and range of each relation Practice #1: Find the domain and range of each relation. Then create a mapping and state whether or not the relation is a function. a) {(–2, 2), (5, 2), (–1, 2), (0, 2), (–4, 2), (2, 2)} b) {(4, 5), (4, 2), (4, –1), (4, 3), (4, 0), (4, –2)} Domain = {__________________} –4, –2, –1, 0, 2, 5 Domain = {__________________} 4 Range = {__________________} 2 Range = {__________________} –2, –1, 0, 2, 3, 5 input output input output –2 –4 –2 –1 –1 2 4 2 2 3 5 5 Yes, this is a function. No, this is not a function.

Practice #2: Graph the examples on the provided set of axes below. a) function b) Not a function {(–2, 2), (5, 2), (–1, 2), (0, 2), (–4, 2), (2, 2)} {(4, 5), (4, 2), (4, –1), (4, 3), (4, 0), (4, –2)} –6 6 y x –6 6 y x What observations can you make from the graphs of the relations? How can we determine whether or not a relation is a function given the graph? If you draw a vertical line through the graph and it hits more than one point at a time, then the relation is not a function.

Vertical Line Test If you pass a ________ line across your graph and it _______ touches the graph at _______ than _____ point at a time, then it is function. vertical never more one If you pass a _______ line across your graph and it touches the graph at ______ than _____ point at a time, then it is ______ function. vertical more one NOT

B) No, most inputs have two outputs. A) Yes C) Yes Examples: The graphs of several relations are shown below. Decide if each is a function. If the relation is not a function, explain why not. a) b) c) B) No, most inputs have two outputs. A) Yes C) Yes d) e) f) D) No, x = -1 has two outputs. F) No, most of the inputs have two outputs. E) Yes

Set-Builder Notation ○ ● ≠  or –   { } efficiently Key Symbols : __________________ is commonly used to __________ represent a set of numbers, especially when we write the domain and range of functions. Set–builder notation efficiently Key Symbols : Represents: { } curly braces, ______________ denotes the set | “_________” such that inequality symbols, indicates the _________ or _________ values in the set minimum >, <, ≥, ≤ maximum ○ _____ point, approaches but not equal to open ● _______ point, can equal to closed not equal ≠ “_________,” indicates that a value is not part of the set “____________” real numbers  or –  negative infinity “_______” or “________________” infinity  “_______________” is an element of

Examples:  Use set–builder notation to write the domain. min max 1) x -3 -2 -1 1 2 3 4 5 6 7 8  {x | } –2 < x ≤ 5 The set of all real numbers x x is greater than −2 and less than or equal to 5 such that max 2) min x -3 -2 -1 1 2 3 4 5 6 7 8  {x | } 1 ≤ x <  The set of all real numbers x such that x is greater than or equal to 1

Examples:  Use set–builder notation to write the domain. min max 3) x -3 -2 -1 1 2 3 4 5 6 7 8  {x | } x   or {x | all real numbers}  or {x | –  < x <  } The set of all real numbers x such that x is an element of the real numbers. min max 4) x -3 -2 -1 1 2 3 4 5 6 7 8  {x | } x  4 The set of all real numbers x such that x is not equal to 4

More on Domain & Range We can determine the domain and range of a relation or function based on the graph itself. Using an index card or something with a straight edge, slowly move the index card from left to right across the graph. Whenever the index card touches the graph of the relation and the x–axis at the same time, highlight the x–axis at that particular spot. This indicates all the possible x–values that are inputs of the relation. Similarly, slowly move the card from down to up across the graph. Whenever the index card touches the graph of the relation and the y–axis at the same time, highlight the y–axis at that particular spot. This indicates all the possible y–values that are outputs of the relation.

Practice: a) b) c) d) e) f) Determine the domain and range for each graph. Use set–builder notation. Practice: a) b) c) d) e) f)

Practice: a) b) c) D: {x | –4 ≤ x <  } D: {x | –3 ≤ x < } Determine the domain and range for each graph. Use set–builder notation. Practice: a) b) c)  D: {x | –4 ≤ x <  }  D: {x | –3 ≤ x < }  D: {x | –3 ≤ x < 3}  R: {y | –1 ≤ y ≤ 4}  R: {y | y  R }  R: {y | –2 ≤ y ≤ 2}

Practice: d) e) f) D: {x | x = –4, –1, 0, 2} D: {x | x  R } Determine the domain and range for each graph. Use set–builder notation. Practice: d) e) f)  D: {x | x = –4, –1, 0, 2}  D: {x | x  R }  D: {x | –2 ≤ x ≤ 4}  R: {y | y = –3, –1, 0, 1}  R: {y | –2 ≤ y < }  R: {y | –2 ≤ y ≤ 4}

Complete the Worksheet http://www.biology.arizona.edu/biomath/tutorials/Notation/SetBuilderNotation.html