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Algebra 2 Relations and Functions Lesson 2-1 Part 1 Goals Goal • To graph relations. • To identify functions. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems. Essential Question Big Idea: Function • What are relations and when is a relation a function? Vocabulary • • • • • Relation Domain Range Function Vertical Line Test Definitions: • Relation – is a pairing of input values with output values. It can be shown as a set of ordered pairs (x,y), where x is an input and y is an output. • Domain – the set of input values for a relation. The x-coordinate of an ordered pair. • Range – the set of output values for a relation. The y-coordinate of an ordered pair. Four Ways to Represent a Relation • • • • Ordered Pairs (input, output) Visually (mapping diagram) Numerically (table of values) Graphical (points or curve on graph) Relation – Ordered Pairs • Set of ordered pairs: {(-2, 3), (0, 0), (2, 3), (4, -1)} – Domain (set of inputs): {-2, 0, 2, 4} – Range (set of outputs): {3, 0, 3, -1} • Set of ordered pairs: {(-1, 1), (-1, -1), (0, 3), (2, 4)} – Domain (set of inputs): {-1, 0, 2} – Range (set of outputs): {1, -1, 3, 4} • Set of ordered pairs: {(-2, 6), (-1, 6), (0, 6), (1, 6), (2, 6)} – Domain (set of inputs): {-2, -1, 0, 1, 2} – Range (set of outputs): {6} Your Turn: Give the domain and range for this relation: {(100,5), (120,5), (140,6), (160,6), (180,12)}. List the set of ordered pairs: {(100, 5), (120, 5), (140, 6), (160, 6), (180, 12)} Domain: {100, 120, 140, 160, 180} The set of x-coordinates. Range: {5, 6, 12} The set of y-coordinates. Relations – Mapping Diagram Relations can be expressed as a correspondence or mapping from one set to another. In the example below the arrows from 1 to 2 indicates that the ordered pair (1, 2) belongs to the relation. Each first component is paired with a second component. x-axis values y-axis values 1 –2 3 2 4 –1 Relations – Mapping Diagram 1. – Example: 1 -3 7 2. 9 2 -5 4 1 -2 – 1. Relation: {(1,2), (-3,9), (7,-5)} • Domain: {1, -3, 7} • Range: {9, 2, -5} – 2. Relation: {(4,-6), (1,3), (1,0), (-2,7)} • Domain: {4, 1, -2} • Range: {3, 0, -6, 7} 3 0 -6 7 Your Turn: State the relation and give it’s domain and range. A 2 B C Relation: {(2, A), (2, B), (2, C)} Domain: {2} Range: {A, B, C} Your Turn: State the relation and give it’s domain and range. 1 5 7 -1 0 2 11 Relation {(1, 0), (5, 2), (7, 2), (-1, 11)} Domain {1, 5, 7, -1} Range {0, 2, 11} Relations – Table of Values Table of Values: Relation: {(-1, -4), (0, -1), (1, 2), (2, 5)} Domain: {-1, 0, 1, 2} Range: {-4, -1, 2, 5} Relations – Table of Values Table of values real world situation. Relation: {(2001, 20.8), (2002, 20.6), (2003, 20.8), (2004, 20.9), (2005, 21.6), (2006, 22.3), (2007, 22.6)}. Domain: {2001, 2002, 2003, 2004, 2005, 2006, 2007}. Range: 20.8, 20.6, 20.8, 20.9, 21.6, 22.3, 22.6}. Your Turn: State the relation and give it’s domain and range. Relation: {(1, 12.5), (2, 16), (4, 22), (6, 24), (9, 25), (12, 26)}. Domain: {1, 2, 4, 6, 9, 12}. Range: {12.5, 16, 22, 24, 25, 26}. Relations – Graphical Example: Relation, individual points on a graph. y The ordered pairs in the relation: (– 3, 0), (– 2, 2), (– 1, 4), (0, 1), (1 – 2), and (2, 1). 4 3 2 1 x 4 3 2 1 2 3 4 1 2 3 4 The domain is the set of all x-coordinates: {– 3, – 2, – 1, 0, 1, 2}. The range is the set of all y-coordinates: {0, 2, 4, 1, – 2, 1}. Relations – Graphical Example: Relation, straight line or curve on a graph. y Relation: 4 2 y x 2. 3 3 2 1 x 4 3 2 1 2 3 4 1 2 3 4 The domain: all real numbers. The range: all real numbers. Relations – Graphical Example: Relation, straight line or curve on a graph. Domain: {-5 ≤ x ≤ 9} Range: {-5 ≤ y ≤ 5} Your Turn: Give the domain and range for the relation shown in the graph. List the set of ordered pairs: {(–2, 2), (–1, 1), (0, 0), (1, –1), (2, –2), (3, –3)} Domain: {–2, –1, 0, 1, 2, 3} The set of x-coordinates. Range: {–3, –2, –1, 0, 1, 2} The set of y-coordinates. Your Turn: Give the domain and range of each relation. y (1, 2) (– 1, 1) x (0, – 1) The domain is the set of x-values which are {– 1, 0, 1, 4}. The range is the set of y-values which are {– 3, – 1, 1, 2}. (4, – 3) 2.3 - 20 Your Turn: Give the domain and range of each relation. y 6 x –4 4 The x-values of the points on the graph include all numbers between –4 and 4, inclusive. The yvalues include all numbers between –6 and 6, inclusive. The domain is {–4≤x≤4}. The range is {–6≤x≤6}. –6 2.3 - 21 Your Turn: Give the domain and range of each relation. y x The arrowheads indicate that the line extends indefinitely left and right, as well as up and down. Therefore, both the domain and the range include all real numbers. 2.3 - 22 Your Turn: Give the domain and range of each relation. y The arrowheads indicate that the line extends indefinitely left and right, as well as upward. The domain is all real numbers. Because there x is at least y-value, –3, the range includes all numbers greater than, or equal to –3 or {y ≥ -3}. 2.3 - 23 Definition: Function A function is a relation in which, for each distinct value of the first component of the ordered pair (x-value), there is exactly one value of the second component (y – value). Example: Function Suppose you are told that a person entered a word into a text message using the numbers 6, 2, 8, and 4 on a cell phone. It would be difficult to determine the word without seeing it because each number can be used to enter three different letters. Number {Number, Letter} {(6, M), (6, N), (6, O)} {(2, A), (2, B), (2, C)} {(8, T), (8, U), (8, V)} {(4, G), (4, H), (4, I)} The numbers 6, 2, 8, and 4 each appear as the first coordinate of three different ordered pairs. Example: Function However, if you are told to enter the word MATH into a text message, you can easily determine that you use the numbers 6, 2, 8, and 4, because each letter appears on only one numbered key. {(M, 6), (A, 2), (T, 8), (H,4)} The first coordinate is different in each ordered pair. A relation in which the first coordinate is never repeated is called a function. In a function, there is only one output for each input, so each element of the domain is mapped to exactly one element in the range. Function: Although a single input in a function cannot be mapped to more than one output, two or more different inputs can be mapped to the same output. 10 1 2 11 3 4 5 6 14 Function: Not a function: The relationship from number to letter is not a function because the domain value 2 is mapped to the range values A, B, and C. Function: The relationship from letter to number is a function because each letter in the domain is mapped to only one number in the range. Example: Decide whether the relation defines a function. F (1,2),( 2,4)(3, 1) Solution Relation F is a function, because for each different x-value there is exactly one y-value. We can show this correspondence as follows. 1, 2, 3 x-values of F 2, 4, 1 y-values of F 2.3 - 29 Example: Decide whether the relation defines a function. G (1,1),(1,2)(1,3)(2,3) Solution As the correspondence shows below, relation G is not a function because one first component corresponds to more than one second component. 1, 2 x-values of G 1, 2, 3 y-values of G 2.3 - 30 Example: Decide whether the relation defines a function. H ( 4,1),( 2,1)( 2,0) Solution In relation H the last two ordered pairs have the same x-value paired with two different y-values, so H is a relation but not a function. Different y-values H ( 4,1), ( 2,1)( 2,0) Not a function Same x-values 2.3 - 31 Example: Give the domain and range of the relation. Tell whether the relation defines a function. 4 6 7 –3 100 200 300 The domain is {4, 6, 7, –3}; the range is {100, 200, 300}. This mapping defines a function. Each x-value corresponds to exactly one y-value. 2.3 - 32 Example: Give the domain and range of the relation. Tell whether the relation defines a function. x –5 0 5 y 2 2 2 This relation is a set of ordered pairs, so the domain is the set of xvalues {–5, 0, 5} and the range is the set of y-values {2}. The table defines a function because each different x-value corresponds to exactly one y-value. 2.3 - 33 Your Turn: Determine whether the relation is a function. x-axis values y-axis values –4 –2 1 0 Not a Function Your Turn: Determine whether the relation is a function. There is only one price for each shoe size. The relation from shoe sizes to price makes is a function. Your Turn: Determine whether the relation is a function. {(10, 5), (20, 5), (30, 5), (60, 100), (90, 100)} Not a Function Your Turn: Determine whether the relation is a function. y 5 4 3 2 1 0 -4 -3 -2 -1 0 1 -1 -2 -3 Function 2 3 4 x Your Turn: Determine whether the relation is a function. Joe Samantha Laura Anna Julie Ian Hilary Chelsea Barbara George Sue Humans Function Mothers Your Turn: Are these functions? { (0, 2), (1, 0), (2, 6), (8, 12) } { (0, 2), (1, 0), (2, 6), (8, 12), (9, 6) } { (3, 2), (1, 0), (2, 6), (8, 12), (3, 5), } { (3, 2), (1, 2), (2, 2), (8, 2), (7, 2) } { (1, 1), (1, 2), (1, 5), (1, -3), (1, -5) } Vertical Line Test Every point on a vertical line has the same xcoordinate, so a vertical line cannot represent a function. If a vertical line passes through more than one point on the graph of a relation, the relation must have more than one point with the same x-coordinate. Therefore the relation is not a function. Example: Use the vertical line test to determine whether each relation graphed is a function. y (1, 2) (– 1, 1) This graph represents a x function. (0, – 1) (4, – 3) 2.3 - 41 Example: Use the vertical line test to determine whether each relation graphed is a function. y 6 x –4 4 This graph fails the vertical line test, since the same x-value corresponds to two different y-values; therefore, it is not a function. –6 2.3 - 42 Example: Use the vertical line test to determine whether each relation graphed is a function. y x This graph represents a function. 2.3 - 43 Example: Use the vertical line test to determine whether each relation graphed is a function. y x This graph represents a function. 2.3 - 44 Your Turn: Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is a function. Any vertical line would pass through only one point on the graph. Your Turn: Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is not a function. A vertical line at x = 1 would pass through (1, 1) and (1, –2). Your Turn: Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is a function. Any vertical line would pass through only one point on the graph. Your Turn: Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is not a function. A vertical line at x = 1 would pass through (1, 2) and (1, –2). Relations Note Graphs that do not represent functions are still relations. Remember that all equations and graphs represent relations and that all relations have a domain and range. Essential Question - Review Big Idea: Function • What are relations and when is a relation a function? • A relation is a set of pairs of input and output values. A function is a relation in which each input corresponds with exactly one output. Assignment: • Section 2-1 part 1, Pg 68 – 69; #1 – 7 all, 8 – 22 even.