CHECK Your Understanding

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1 CHECK Your Understanding. State the domain and range of each relation. Then determine whether the relation is a function, and justif our answer.. a) e) 5(, ), (, 9), (, 7), (, 5), (, ) 5 5 f) 55. State the domain and range of each relation. Then determine whether the relation is a function, and justif our answer. a) 5( ) 5 e) cos f) 5 sin PRACTISING. Determine whether each relation is a function, and state its domain and range. a) e) 5 7 5(, ), (, ), (5, ), 5(, 5), (, ), (, 7), f) (, ) (, ) 5(, ), (, ), (, ), (, ) NEL Chapter

2 . Determine whether each relation is a function, and state its domain K and range. a) 5 5 e) 5 f) f () 5 5. Determine the equations that describe the following function rules: a) The input is less than the output. The output is 5 less than the input multiplied b. Subtract from the input and then multipl b to find the output. The sum of the input and output is 5. w l closet wall wall. Martin wants to build an additional closet in a corner of his bedroom. Because the closet will be in a corner, onl two new walls need to be built. The total length of the two new walls must be m. Martin wants the length of the closet to be twice as long as the width, as shown in the diagram. a) Eplain wh l 5 w. Let the function f (l ) be the sum of the length and the width. Find the equation for f (l ). Graph 5 f (l ). Find the desired length and width. 7. The following table gives Tina s height above the ground while riding a A Ferris wheel, in relation to the time she was riding it. Time (s) Height (m) a) Draw a graph of the relation, using time as the independent variable and height as the dependent variable. What is the domain? What is the range? Is this relation a function? Justif our answer. e) Another student sketched a graph, but used height as the independent variable. What does this graph look like? f) Is the relation in part e) a function? Justif our answer.. Functions NEL

3 . Consider what happens to a relation when the coordinates of all its ordered pairs are switched. a) Give an eample of a function that is still a function when its coordinates are switched. Give an eample of a function that is no longer a function when its coordinates are switched. Give an eample of a relation that is not a function, but becomes a function when its coordinates are switched. 9. Eplain wh a relation that fails the vertical line test is not a function.. Consider the relation between and that consists of all points (, ) such that the distance from (, ) to the origin is 5. a) Is (, ) in the relation? Eplain. Is (, 5) in the relation? Eplain. Is the relation a function? Eplain... The table below lists all the ordered pairs that belong to the function g(). 5 g() 7 9 a) Determine an equation for g(). Does g() g() 5 g( )? Eplain.. The factors of are,, and. The sum of the factors is T 5 7. The sum of the factors is called the sigma function. Therefore, f () 5 7. a) Find f (), f (7), and f (). Is f () 5 f () f ()? Is f (5) 5 f () f (5)? Are there others that will work?. Make a concept map to show what ou have learned about functions. C Put FUNCTION in the centre of our concept map, and include the following words: algebraic dependent variable domain function notation Etending graphical independent variable mapping numerical range vertical line test. Consider the relations 5 5 and 5 "5. Draw the graphs of these relations, and determine whether each relation is a function. State the domain and range of each relation. 5. You alread know that is a function of if and onl if the graph passes the vertical line test. When is a function of? Eplain. Communication Tip A concept map is a tpe of web diagram used for eploring knowledge and gathering and sharing information. A concept map consists of cells that contain a concept, item, or question and links. The links are labelled and denote direction with an arrow smbol. The labelled links eplain the relationship between the cells. The arrow describes the direction of the relationship and reads like a sentence. NEL Chapter

4 Answers Chapter Getting Started, p.. a) 5 a 5a. a) ( )( ) (5 )( ) ( )( ) (a ( ). a) horizontal translation units to the right, vertical translation units up; horizontal translation unit to the right, vertical translation units up; horizontal stretch b a factor of, vertical stretch b a factor of, reflection across the -ais; horizontal compression b a factor of vertical stretch b a factor of,, reflection across the -ais; a) D 5 5PR # #, f) R 5 5PR # # D 5 5PR, R 5 5 PR $9 D 5 5PR, R 5 5 PR D 5 5PR, R 5 5 PR # $ e) D 5 5PR, R 5 5 PR. 5. a) This is not a function; it does not pass the vertical line test. This is a function; for each -value, there is eactl one corresponding -value. This is not a function; for each -value greater than, there are two corresponding -values. This is a function; for each -value, there is eactl one corresponding -value. e) This is a function; for each -value, there is eactl one corresponding -value.. a) about.7 7. If a relation is represented b a set of ordered pairs, a table, or an arrow diagram, one can determine if the relation is a function b checking that each value of the independent variable is paired with no more than one value of the dependent variable. If a relation is represented using a graph or scatter plot, the vertical line test can be used to determine if the relation is a function. A relation ma also be represented b a description/rule or b using function notation or an equation. In these cases, one can use reasoning to determine if there is more than one value of the dependent variable paired with an value of the independent variable. Lesson., pp.. a) D 5 5PR; R 5 5PR # #; This is a function because it passes the vertical line test. D 5 5PR # # 7; R 5 5 PR # # ; This is a function because it passes the vertical line test. D 5 5,,, ; R 5 55,, 7, 9, ; This is not a function because is sent to more than one element D 5 5PR; R 5 5PR; This is a function because ever element in the domain produces eactl one element e) D 5 5,,, ; R 5 5,,, ; This is a function because ever element of the domain is sent to eactl one element D 5 5PR; R 5 5PR # ; This is a function because ever element in the domain produces eactl one element. a) D 5 5PR; R 5 5 PR #; This is a function because ever element in the domain produces eactl one element D 5 5PR ; R 5 5 PR ; This is a function because ever element in the domain produces eactl one element in the range. D 5 5PR; R 5 5 PR. ; This is a function because ever element in the domain produces eactl one element D 5 5PR; R 5 5 PR # # ; This is a function because ever element in the domain produces eactl one element e) D 5 5PR # # ; R 5 5PR # # ; This is not a function because (, ) and (, ) are both in the relation. f) D 5 5PR; R 5 5PR # # ; This is a function because ever element in the domain produces eactl one element. a) function; D 5 5,, 5, 7; R 5 5,, function; D 5 5,,, 5; R 5 5,, function; D 5 5,,, ; R 5 5, not a function; D 5 5,, ; R 5 5,, 5, 7 e) not a function; D 5 5,, ; R 5 5,,, f) function; D 5 5,,, ; R 5 5,,,. a) function; D 5 5PR; R 5 5 PR $. not a function; D 5 5PR $ ; R 5 5 PR function; D 5 5PR; R 5 5 PR $.5 not a function; D 5 5PR $ ; R 5 5 PR e) function; D 5 5PR ; R 5 5 PR f) function; D 5 5PR; R 5 5 PR 5. a) 5 5 ( ) Answers NEL

5 . a). a) The length is twice the width. f () 5 ; f (7) 5 ; f () 5 5 f (l ) 5 l Yes, f (5) 5 f () f (5) Yes, f () 5 f () f () f(b) Yes, there are others that will work. f (a) f ( 5 f (a whenever a and b have no common factors other than.. Answers ma var. For eample: B independent numerical variable length 5 m; width 5 m 7. a) Time (s) D 5 5,,,,,,,,,,,, R 5 5, 5, It is a function because it passes the vertical line test. e) Height (m) Time (s) Height (m) f) It is not a function because (5, ) and (5, ) are both in the relation.. a) 5(, ), (, ), (5, ) 5(, ), (, ), (5, ) 5(, ), (, ), (5, ) 9. If a vertical line passes through a function and hits two points, those two points have identical -coordinates and different -coordinates. This means that one -coordinate is sent to two different elements in the range, violating the definition of function.. a) Yes, because the distance from (, ) to (, ) is 5. No, because the distance from (, 5) to (, ) is not 5. No, because (, ) and (, ) are both in the relation.. a) g() 5 g() g() g( ) 5 g() 5 So, g() g() g( ).. domain range dependent variable The first is not a function because it fails the vertical line test: D 5 5PR 5 # # 5; R 5 5PR 5 # # 5. The second is a function because it passes the vertical line test: D 5 5PR 5 # # 5; R 5 5PR # # is a function of if the graph passes the horizontal line test. This occurs when an horizontal line hits the graph at most once. Lesson., p. FUNCTION function notation. 5,, 5,, 5. a) e) 5 f). a). $ # 5. a) mapping algebraic graphical vertical line test The absolute value of a number is alwas greater than or equal to. There are no solutions to this inequalit. 5. a) # $.,. a) The graphs are the same. Answers ma var. For eample, 5( ), so the are negatives of each other and have the same absolute value. 7. a). When the number ou are adding or subtracting is inside the absolute value signs, it moves the function to the left (when adding) or to the right (when subtracting) of the origin. When the number ou are Answers NEL Answers