UNIT THREE FUNCTIONS MATH 611B 15 HOURS

Transcription

1 UNIT THREE FUNCTIONS MATH 611B 15 HOURS Revised Jan 9, 03 51

2 SCO: By the end of Grade 1, students will be epected to: C1 gather data, plot the data using appropriate scales, and demonstrate an understanding of independent and dependent variables and domain and range C eplore and apply functional relationships and notation, both formally and informally Elaborations - Instructional Strategies/Suggestions Relations and Functions (1.1) Much of the material in this unit will be a review from previous courses (consult the Math 51B teacher in your school). The topics that are new will be highlighted and labelled (new material) and will account for a significant portion of the time allotted for this unit. Student groups should do the review eercises in the Suggested Resources. Student groups should read and discuss p.6-8 in the tet. Key concepts that should come out in the discussion are: < domain and range < independent and dependent variable < functional notation < input and output < ( see p.7-8) When a function is defined with a formula, and the domain is not stated eplicitly or restricted by contet, the domain is assumed to be the largest set of -values for which the formula gives real y-values and is called the natural domain. If we want to restrict the domain, we must say so. An independent variable is a variable that a researcher is able to control and is the horizontal ais when the data is graphed. A dependent variable is a variable that a researcher believes is affected by changes in an independent variable. The measurements recorded are those of the dependent variable and are graphed on the vertical ais. The independent variable is the input (domain). For instance, if you want to do an internet search on the input is what you type into the search window. The dependent variable is the output (range). The internet search results are the output. A relation is a rule that produces one or more output numbers for every valid input number. A function is a rule that gives a single output number for every valid input number. 5

3 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Relations and Functions (1.1) Communication Eplain the differences between relations and functions to the other members of your group. Journal Write to eplain the terms independent and dependent variables and why they might be called such in a scientific eperiment. Pencil/Paper For the function f() = + 3! 6 determine: * should be emphasized a) f () f) * f ( a + ) b) f (3 / 5) g)* f ( + h) f (3 t) f ( a 5) c) h)* d) f ( 5) i) * f ( h) e)* f (t 1) j)* f ( a + 1) Relations and Functions (1.1) Calculus & Advanced Functions Prerequisite Skills p #1(d),(d),3-5,9,18(c),(g), (h) Prerequisite Skills p.4 #1,4,5,6(d),7(c),8(c),9(c),10(b), 13(a),(d),16 Functions p.19 # 6,7 Interval Notation p. 18 # 1 See Relations and Functions Worksheet (1.1)[contains above problems] Group discussion Eplain the differences between a relation and a function to one of your group members. Research Find five real world eamples of functions. Sketch graphs of each. Project Research and describe five eamples of relations in everyday life. Sketch a graph of each relationship. Be prepared to share your eamples with the class. Pencil/Paper Write interval notation for each of the following: a)! < # 5 b) 1 < < 4 c) >!3 Journal Write to eplain the concept of interval notation. 53

4 Relations and Functions Worksheet (1.1) Page 40, Calculus tet 1.(d) Epand and simplify ( w 3) ( w + 5)( w 4)..(d) Simplify Factor. a) t + 9t b) w 196 c) d) y 3y 18 e) f) w 38w g) t 98t 99 h) s + 90s Factor. a) b) 3y 11y 4 c) 4w + 9w + d) 10a a 3 e) 6t + 7t + f) Simplify. State any restrictions on the variable. a) b) t t 6 c) t t 8 d) a 9 a + 0 a + a 30 54

5 9. Solve using the quadratic formula. Epress solutions both in eact form and as approimations to the nearest tenth. a) 5 = 0 b) y + 3y + 1 = 0 c) w + w = 4 d) 0 = 4t + t Solve. Graph the solution on a number line. (c) ( 3) 4 3( 1) + (g) (h) > 6 page 4, Calculus tet 1. If f() = 3 +! +3, find a) f(0) b) f(1) c) f(1/) d f(!1) e) f(0.1) f) f(a) g) f() h) f(!) 4. If f ( ) =, find a) f(0) b) f() c) f(!) d) f(5) 3 5. For each graph, evaluate f at the following values of. Then state the domain and range of f. i) f() ii) f(!) iii) f(0) iv) f(4) 55

6 6.(d) Find the slope and y-intercept (! 3)! y = 3. 7.(c) Find an equation of the line with slope! and y-intercept 4. 8.(c) Find an equation of the line with slope passing through (!, 5). 9.(c) Find an equation of the line with -intercept! and y-intercept (b) Find an equation of the line passing through (5, 3) and (!3, 1). 13. Write each quadratic in the form y = a(! p) + q. a) y = d) y = Describe the transformations applied to the graph of g(). 1 a) g ( 5) b) 3g( ) c) g( ) 9 d) g( 4( + 1)) 3 3 Functions p.19, Calculus tet 6. For each function, find i) f() ii) f(!) iii) f(1/) iv) f(1/3) v) f(k) vi) f(1! k) a) f() = 1! b) f() = 7. For each function, find and simplify i) f(3) ii) f(!3) iii) f(1/3) iv) f(1/4) v) f(1/k) vi) f F HG k 1 + k I K J a) f ( ) = 1 b) f ( ) = 1 Interval notation p.18, Calculus tet 1. Use interval notation to epress the set of real values described by each inequality. Illustrate each interval on the real number line. a) b) 4 13 c) 4 < < 1 d) 0 < < 4 e) < f) > 1 g) 1 h) 0 56

7 Relations and Functions Worksheet (1.1) Etra Problems 1. Determine the domain and range and state which of the following are functions: a) {(!3,1),(!,4),(1,1),(3,4)} b) {(!1,1),(0,4),(!1,3),(4,)} c) {(!,0),(!1,0),(0,0),(1,5),(,0)}. Eplain the differences between a relation and a function to one of your group members. Graph the three problems in #1 above and use the graphs to strengthen your arguments. Eplain why you believe the ones that you have chosen to be functions are in fact functions. 3. Find the equation of the perpendicular bisector of the line segment A (, 5) and B (6, 11). 4. Find the equation of the line going through point A (!, 4) and parallel to the line defined by the equation + 5y = Find the equation of the line going through point A (4,!1) and perpendicular to the line defined by the equation! + 4y = Determine the domain and range and write in interval notation. State which of the following are functions: a) b) 57

8 c) d) e) f) 58

9 g) h) i) 59

10 1. a) D = {!3,!, 1, 3} R = {1,4} function Answers for Relations & Functions Worksheet (1.1) b) D = {!1, 0, 4} R = {1,,3,4} not a function c) D = {!,!1, 0, 1, } R = {0, 5} function y! 3 = y! 16 = y! 7 = 0 6. a) D = [!7, 4] R = [!6, 4] function b) D = [!6, 6] R = [!5, 6] function c) D = (!7, 7) R = (!3, 3] function d) D = [!3, 5] R = (!5, 5] not a function e) D = [!6, 3) R = (!4, 0) not a function f) D = (!4, 4) R = (!4, 4) function g) D = (!4,!1.5] c [!1, ] c [3, 4) function R = [!, 4) h) D = (!6, ] R = (!7, 1] not a function i) D = [0, 4) R = (!4, 4) not a function 60

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12 SCO: By the end of Grade 1, students will be epected to: C eplore and apply functional relationships and notation, both formally and informally Elaborations - Instructional Strategies/Suggestions Even, Odd Functions & Symmetry (1.1) Algebra definition: A function is even iff f(!) = f() for all in the domain of f. A function is odd iff f(!) =!f() for all in the domain of f. Algebra: Students should be able to show algebraically whether a function is even, odd or neither. Geometric Properties of Even and Odd Functions: A function is even if its graph is symmetric with respect to the y-ais. (This means that when a Reflect-View is placed on the y-ais the left half of the graph is reflected onto the right half.) A function is odd if its graph is symmetric with respect to the origin. (This means that the left half of the plane looks like the mirror image of the right half only upside down.) Students should be able to recognize from the graph of a function whether that function is even, odd or neither. Note to Teachers: Students should be able to identify even and odd functions and therefore symmetries before they graph the functions. In this way students may know what type of graph to epect before they start to plot points. 6

13 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Even, Odd Functions & Symmetry (1.1) Journal Discuss the symmetry of the graph of each function below: a) 3 f ( ) = 5 b) f ( ) = + 3 c) f ( ) = Even, Odd Functions & Symmetry (1.1) Even, Odd Functions & Symmetry Worksheet (1.1) 63

14 Even, Odd Functions & Symmetry Worksheet (1.1) Etra Problems 1.Determine whether each function is even, odd or neither. Then state the domain of each function in interval notation. a) f b) f ( ) = cos ( ) = + 3 page 18. Calculus tet 10. i) f ( ) = j) f ( ) = Eplain why the following functions are even: a) g( ) = + b) n ( ) = 3 c) t 4 ( ) =

15 Answers for Even, Odd Functions & Symmetry Worksheet (1.1) - Etra problems 1. a) neither b) even 65

16 SCO: By the end of Grade 1, students will be epected to: C eplore and apply functional relationships and notation, both formally and informally Elaborations - Instructional Strategies/Suggestions (New material) Piecewise functions are a class of functions such that the formula used depends on the part of the domain it refers to. Separate formulas are generally required to define each part of a piecewise function. Piecewise Functions: < absolute value function < step function < general function Student groups should read and discuss p.1-14 in the tet. Student groups should be able to graph and become familiar with the names of various functions and determine the natural domain and range of each. Use technology to visualize the various types of graphs and discontinuities, if they eist. Students should use the vertical line test (VLT) to verify that the graphs are functions. A technology eample could be: 3, (,0] 3 f() = 3, (0,] 8, (3, ) 66

17 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Piecewise Functions (1.1) Pencil/Paper/Technology Sketch the graph of each of the following. Discuss the symmetry of each graph and determine whether each are even or odd functions or neither. R S, (, ) a) f() = T 1 + 1, ( 1, ) b) f() = c) f() = R S T R S T, (, ) 4, [, ) + 3, [, ), (, 3] 1, ( 3, ) Piecewise Functions (1.1) Piecewise Function Worksheet (1.1) d) f ( ) = 4 e) f ( ) = f) f ( ) = 1 g) f ( ) =

18 Piecewise Function Worksheet (1.1) p.19 Calculus tet 5. Determine whether each function is even or odd or neither. Then graph the function on the domain (, ). a) f ( ) = b) g( ) = + 1 c) h ( ) = 3 Etra Problems Sketch the graph of each of the following. 1. question (a) on p.67. question (b) on p question (c) on p f ( ) = 5. f ( ) 6. f ( ) = 7. f ( ) = = R S T R S + 3 (, ] (, 3] 10 (, 3 ) R S (, ) T [ 0, ) = T R S T 6 (, ] 3 (, ) 4 [, ) 8. f ( ) R S T (, ] = (, ) 9. f ( ) = (, 0) + [ 0, ) Sketch graphs for each of the following. Discuss the symmetries and whether the functions are even, odd 68

19 or neither. 10. f ( ) = f ( ) = 4 1. f ( ) = f ( ) = 14. f ( ) =

20 Answers for Piecewise Function Worksheet (1.1) - Etra problems

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28 SCO: By the end of Grade 1, students will be epected to: C6 develop and apply strategies for solving problems Elaborations - Instructional Strategies/Suggestions Roots of Polynomial Equations (.5) The topics discussed in this section were covered in Math 51B Unit 6. Student groups should do a brief review of synthetic division. An eample is shown below. E. ( ) ( ) Use the coefficients of the polynomial. Include the coefficient 0 for any missing terms. Divide by the constant of the divisor. So the quotient is 4! 3! 6 remainder of 0. Student groups should also do a brief review of Remainder Theorem (.3), Rational Root Theorem (.4) and Factor Theorem (.4). This should allow students to solve for roots of polynomial equations. 78

29 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Roots of Polynomial Equations (.5) Pencil/Paper Use the remainder theorem to determine the remainder for each division. a) ( + + 4) (! ) b) (y 3! 8) (y + ) Roots of Polynomial Equations (.5) Features of Polynomial Graphs Investigation p Roots of Polynomial Equations (.5) Worksheet Pencil/Paper Factor completely. a) k 3 + 6k!7k! 60 b) Pencil/Paper Solve by factoring. a) d 3! 3d! 1d! 7 = 0 b) 6y 3 + 9y! 1 = 3y 79

30 Roots of Polynomial Equations Worksheet (.5) p.59 Calculus tet 5. Use the remainder theorem to determine the remainder for each division. a) ( + + 4) ( ) b) ( 4n + 7n 5) ( n + 3) 3 c) ( ) ( 1) 3 d) ( ) ( + 1) 6. Use the remainder theorem to determine the remainder for each division. a) ( ) ( 3) b) ( 6a + 5a 4) ( 3a + 4) 3 c) ( ) ( 1) 3 d) ( y + y 6y + 3) ( y + 1) p.68 Calculus tet 6.Factor completely. 3 a) b) c) d) Factor. a) b) 4y 7y 3 3 c) d) p.79 Calculus tet 3. Solve and check. 3 a) = 0 3 c) t + t 7t + 4 = 0 3 e) a 4a + a + 6 = 0 6. Solve by factoring. 3 a) = 0 80

31 3 d) d 3d 1d 7 = 0 f) =

32 SCO: By the end of Grade 1, students will be epected to: C46 investigate and articulate how various changes in the parameters of an equation affect graphs Elaborations - Instructional Strategies/Suggestions Polynomial Functions & Inequalities (.6) Check with Math 51B teacher on coverage. This topic was covered in Math 51B Unit 6. See p in the Math 51B guide for a description of what was covered in this grade 11 course. Consequently although this should be a review for the students time should be allocated for a worthwhile discussion of this topic. What may be new to students is the treatment of leading coefficients and end behaviours of polynomial functions as summarized below. Graphs of above four cases: 1) ) f() = f() =! 3! ) 4) f() = 4! 3 3! + 3 f() =! 4!

33 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Polynomial Functions & Inequalities (.6) Pencil/Paper/Technology Use the graph to determine the domain and range, the approimate coordinates of any local maimums or minimums, and the y intercept of each of the following polynomial functions. a) b) Polynomial Functions & Inequalities (.6) Polynomial Functions & Inequalities Worksheet (.6) Etension: A brief review of Rational Inequalities from Math 51B Unit 6 (see 51B guide). c) d) Pencil/Paper/Technology Graph each function and determine any real zeros, the domain and range, the y-intercept, any local maimums or minimums, any symmetry, and the end behaviour. a) f() =! b) f() = ! 4! 7 + Group Activity Solve each inequality and graph the solution set a) 3! 3! 3 + < 0 b) ! 7! 10 > 0 83

34 Solving Polynomial and Rational Inequalities Polynomial Inequalities Invite students to read pages in the Math 51B tet, then ask for their ideas on what the terms critical numbers and test intervals mean. Allow students to attempt a few quadratic inequalities in the Suggested Resources to develop the algorithms: < critical numbers are the zeros of the polynomial function < test intervals are the regions between the critical numbers E: Solve!! 6 < 0 Factoring yields: (! 3)( + ) < 0 Therefore the critical numbers are! and 3. The test intervals are: Using a table as shown can be found: below the solution From the table it can be determined that the centre region is the solution. Invite students to read pages in the Math 51B tet Challenge students to solve this type of inequality: < factor the numerator and denominator (if possible): < determine the critical numbers < determine the value(s) that make the denominator 0 < use test intervals to find the solution(use the numbers from parts two and three above. 15 E: Solve: 0 1 The critical numbers are!3 and 5. The value for which the denominator is zero is 1. Therefore the test intervals are: 84

35 By using a table as shown below the solution can be found. From the table we can see that regions 1 and 3 are the solution. 85

36 Polynomial Functions & Inequalities (.6) p.9 Calculus tet 1. Use the graph to determine the domain and range, the approimate coordinates of any local maimums or minimums, and the y-intercept of each polynomial function. 86

37 . Use the graph to identify the real zeros of the polynomial function, and the intervals where f ( ) 0 and where f ( ) < 0. Estimate zeros to the nearest tenth, if necessary. c) d) 87

38 3. Graph each function and determine i) any real zeros, to the nearest tenth, if necessary ii) the domain and range iii) the y-intercept iv) the approimate coordinates of any local maimums or minimums v) any symmetry a) f ( ) = ( 4) 3 c) y = 3 d) k( ) = e) y = + 6 g) f ( ) = ( 4)( + 1)( + 1) h) f ( ) = ( )( 4)( 6) 5. Solve each inequality using the method of your choice. Graph each solution on a number line. c) 3 10 e) ( )( ) 0 f) ( 1)( + 1)( ) 0 3 h) i) 3 > 6 Etra Problems 6. Solve each inequality using the method of your choice. Graph each solution on a number line. a) b) > 0 c) d) e) > 0 88

39 7. Match the following graphs with the proper equations. a) b) c) d) e) f) a) f ( ) = ( + )( )( 1) b) f ( ) = ( 1)( 3)( + ) c) f ( ) = ( )( + 1)( 3) d) f ( ) = ( + 4)( + 1)( 1) e) f ( ) = ( + 5) f) f ( ) = ( 1)( + 3)( + 3) 89

40 Solutions for Etra problems. 6. a) 3 1 b) 3 < < 1, > c) < 1, 4 d) 4, > 3 e) < < 1, > 4 7. Graph a) is equation (d) Graph b) is equation (e) Graph c) is equation (a) Graph d) is equation (c) Graph e) is equation (f) Graph f) is equation (b) 90

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42 SCO: By the end of Grade 1, students will be epected to: C46 investigate and articulate how various changes in the parameters of an equation affect graphs Elaborations - Instructional Strategies/Suggestions (New material) Invite student groups to read and discuss p.414 and do the Investigations on p Students should also eamine eamples 1, and 3. The concept of limit is not considered at this time. y ca p = + b < c vertical stretch < p horizontal translation < b vertical translation There are many eamples of eponential functions in real life. In one such eample eponential functions are used to model tornados using the Fujita scale. see: w.tornadoproject.com 9

43 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Eponential Functions (7.1) Eponential Functions (7.1) Puzzle Activity Eplain to the class through a number of graphed eamples the affect that the coefficient c has on the eponential equation y = ca. Eponential Functions Worksheet (7.1) Puzzle Activity Eplain to the class through a number of graphed eamples the affect that the coefficient b has on the eponential equation y = a + b Puzzle Activity Eplain to the class through a number of graphed eamples the affect that the coefficient p has on the eponential equation y = a p Each group should teach the effect of their coefficient at the board so that once all groups have presented the puzzle will be complete. Group Presentation Each group is to design an eponential function containing the coefficients c, b and p and eplain how the values of each of their coefficients affect the basic eponential function. 93

44 Eponential Functions Worksheet (7.1) p.419 Calculus tet 1. a) Eplain the similarities and differences among the graphs of y =, y = 6 and y = 9. In your eplanation, pay attention to the y-intercepts and the limits as approaches positive and negative infinity. b) Repeat part a) for the graphs of y =, and. H G 1I K J y = H G 1 I K J y = 6 H G 1I K J 9. Graph each pair of functions on the same set of aes. First, use a table of values to graph f(), and the, use your graph of f() to graph g(). Check your work using graphing technology. a) f ( ) = 3 and g b) f ( ) = 5 and g c) f ( ) = 6 and g d) f ( ) = 10 and g F ( ) = H G 1 I K J 3 F ( ) = H G 1 I K J 5 F ( ) = H G 1 I K J 6 F ( ) = H G 1 I K J a) Use a table of values to graph f ( ) = 1 b) Describe how to graph g ( ) ( ) and using the graph of 3 h ( ) = ( 3 ) f ( ) =. = F H G I K J 1 c) Graph g ( ) ( ) and on the same set of aes as f(). Confirm your 3 h ( ) = ( 3 ) results with graphing technology. d) State the domain and range of each function. = F H G I K J 94

45 4. Use the given graphs to sketch a graph of each of the following functions without using a table of values or technology. State the y-intercept, domain, range and equation of the asymptote of each function.. a) f ( ) = b) g( ) = + 3 h) y + 3 F = H G I () K J For each function, i) state the y-intercept a) f ( ) = 3 + ii) state the domain and range iii) state the equation of the asymptote iv) draw a graph of the function c) h ( ) = ( 7 ) 5 e) f ( ) = F H G I K J

46 SCO: By the end of Grade 1, students will be epected to: C46 investigate and articulate how various changes in the parameters of an equation affect graphs Elaborations - Instructional Strategies/Suggestions (New material) Invite student groups to read and discuss p.4-44(top of page). Challenge student groups to do the Investigation on p.4. Students should then read p An investigation of the general logarithmic function and the effects that each coefficient has on the basic logarithmic function would be in order. y = clog a( p) + b < a horizontal stretch < c vertical stretch < p horizontal translation < b vertical translation Logarithmic functions occur in real life, an eample is shown below. Eample: A recent sociological study showed that the average walking speed v, in metres per second, of a person living in a city with population n, in thousands, is given by. v = 05. log n Find the average walking speeds in Charlottetown, Halifa and Toronto. This shows the average walking speed for a city of 500,000 people. The logarithmic function is in fact the slowest increasing function of all functions. 96

47 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Logarithmic Functions (7.) Logarithmic Functions (7.) Technology Use the TI-83 to graph y = and y = log on the same screen. State the domain and range of each function. Logarithmic Function Worksheet (7.) Pencil/Paper Graph each function using transformations. State the domain and range and the equation of the vertical asymptote. a) f ( ) = log ( ) b) f ( ) = log( ) + 3 c) f ( ) = log( 1) + 3 d) f( ) = 3 log ( 3) 4 97

48 Logarithmic Functions (7.) p.48 Calculus tet 3. Use graphing technology to graph each pair of functions on the same set of aes. State the domain and range of each function. a) y = and y = log 4 4 b) y = and y = log 8 8 c) f ( ) = and f ( ) = log 4. Determine the equation of the inverse of each function. Graph each equation and its inverse. a) y = 3 b) y = log 4 5. Graph each function using transformations. State the domain, the range and the equation of the vertical asymptote. a) f ( ) = log4 b) y = log ( ) 3 f) y = log 3( + 1) 6. Without graphing, determine the domain of each function. 3 a) f ( ) = log ( 5 ) b) y = log ( 7 ) 3 c) y = log 3( 1) d) y = log 10( 4) 7 98

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50 SCO: By the end of Grade 1, students will be epected to: B1 model(with concrete materials and pictorial representations) and epress the relationship between arithmetic operations and operations on functions Elaborations - Instructional Strategies/Suggestions (New material) Challenge student groups to research operations with functions on the internet, bring their findings to class and compare their findings with other members in their group. Rules for operations with functions are: sum f + g is defined as ( f + g)( ) = f ( ) + g( ) difference f g is ( f g)( ) = f ( ) g( ) product fg is ( fg)( ) = f ( ) g( ) The domain of these three functions is the intersection of domain f and domain g. f f f() quotient is () =, g() 0 g g g () The domain of the f/g function is the intersection of the domain of f and the domain of g ecluding any value of for which g() = 0. Operations with functions can be demonstrated very nicely on the TI-83: Note to Teachers: The only operation to be careful with is multiplication. Represent it as Y 1 * Y on the TI-83. The representation Y 1 (Y ) is used for composition in the net section. 100

51 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Operations with Functions Pencil/Paper Let f = { (1,3),(,8),(3,6),(5,9)} and g = {(1,6),(,11),(3,0),(4,1)}. Find each of the following and determine the domain of each: a) f + g b) f g c) f/g Operations with Functions Operations with Functions Worksheet Pencil/Paper Let f ( ) = + and g ( ) = 3 the following and determine the domain of each: a) f + g b) g/f c) g f d) f! g Find each of 101

52 Operations with Functions Worksheet (3.7) Etra problems 1. Let f = { (1,3),(,8),(3,6),(5,9)} and g = {(1,6),(,11),(3,0),(4,1)}. Find and state the domain of each: a) f + g b) f g c) f/g Solution: a) The domain of f + g is the intersection of f and g {1,,3}. (f + g)(1) = f(1) + g(1) = = 9 or (1,9) (f + g)() = f() + g() = = 19 or (,19) (f + g)(3) = f(3) + g(3) = = 6 or (3,6) Thus f + g = {(1,9),(,19),(3,6)} b) The domain of fa g is the intersection of f and g {1,,3}. (f g)(1) = f(1) A g(1) = 3 A 6 = 18 or (1,18) (f g)() = f() A g() = 8 A 11 = 88 or (,88) (f g)(3) = f(3) A g(3) = 6 A 0 = 0 or (3,0) Thus f g = {(1,18),(,88),(3,0)} c) The domain of f/g is {1,} ; 3 is not included because g(3) = 0. (f/g)(1) = f(1)/g(1) = 3/6 = ½ or (1,1/) (f/g)() = f()/g() = 8/11 or (,8/11) Thus f/g = {(1,1/),(,8/11)} f and g ( ) = 3 ( ) = + 5. Let. Determine and simplify. a) (f + g)(4) b) (f! g)() c) (f g)(0) d) (f/g)(9) f ( ) = and g( ) = and h( ) = 1 3.Let and state the domain of each: a) f + g b) g/f c) g h d) g! h. Find each of the following 10

53 4. State whether the following are true or false; a) If f = {(,4)} and g = {(1,5)}, then f + g = {(3,9)}. b) If f ={(1,6),(9,5)} and g = {(1,3),(9,0)}, then f/g = {(1,)}. c) If f = {(1,6),(9,5)} and g = {(1,3),(9,0)}, then f g = {(1,18),(9,0)}. d) If f() = + and g() =! 3, then (f g)(5) = Let f() =! 3 and g() =!. Find and simplify each epression: a) (f + g)() b) (g + f)(3) c) (f! g)(!) d) (g! f)(!6) e) (f g)(!1) f) (g f)(0) g) (f/g)(4) h) (g/f)(4) i) (f + g)(a) 6. Let f = {(!3,1),(0,4),(,0)}, g = {(!3,),(1,),(,6),(4,0)} and h = {(,4),(1,0)}. Find each function. a) f + g b) f! h c) g/f d) f/g e) f g f) g! h 1 f( ) =, g( ) = 4 and h( ) = 7. Let. Find an equation defining each function below and state the domain of the function: a) f + g b) f + h c) g h d) g/f e) f/g f) f! h g) g/h h) f h y =, y = 1 and y = y + y 8. Enter into the graphing calculator. a) Graph y 3 in [!3,3] by [!1,3]. b) Compare the domain of the graph of y 3 with the domains of the graphs of y 1 and y. c) Replace y 3 by y 1! y, y! y 1, y 1 y, y 1 / y and y / y 1 and repeat the comparison of part (b). d) Make a conjecture about the domains of sums, differences, products and quotients of functions. 103

54 9. Shown below are the graphs of two functions. State the domain of f + g and sketch the graph of f + g on the same grid. a) b) 104

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56 e) f) 106

57 g) h) 107

58 Answers for Operations with Functions Worksheet (3.7) - Etra problems 1. a) f + g = {(1,9),(,19),(3,6)} D = {1,,3} b) f g = {(1,18),(,88),(3,0)} D = {1,,3} c) f/g = {(1,1/),(,8/11)} D = {1,}. a) (f + g)(4) = 5 b) ( f g)( ) = 3 7 c) (f g)(0) =!10 d) (f/g)(9) = 7/86 3. a) f + g = D = [ 0, ) b) g f = D = ( 0, ) c) g h = 3 1 D = (, ) d) g h = + D = (, ) 4. a) true b) true c) true d) true 5. a) ( f + g)( ) = 1 f) ( gf)( 0) = 0 b) ( g + f)( 3) = 6 g) c) ( f g)( ) = 11 h) F fi ( ) = HG gkj 4 F g HG I fk J ( ) = d) ( g f)( 6) = 51 i) ( f + g)( a) = a e) ( fg)( 1) = a) f + g = {( 33, ), ( 6, )} b) f h = {(, 4)} c) g f = {( 3, )} d) f g = {( 3, ), ( 0, )} e) f g = {( 3, ), ( 0, )} f) g h = {( 1, ), (, )} 108

59 7. a) f + g = + 4 [ 0, ) b) f + h = + 1 [ 0, ), c) g h = 4 (, ) (, ) d) g 4 = f ( 0, ) e) f = g 4 [ 04, ) ( 4, ) f) f h = 1 [ 0, ), g) g h = (, ) (, ) h) f h = [ 0, ), 9. a) D = [0,3] b) D = [!, ] 109

60 = [0.5,3] c) D 110

61 d) D = [!1, 1] e) D = [1,] 111

62 f) D = [-,1] g) D = [, 4] 11

63 h) D = [-1,1] 113

64 SCO: By the end of Grade 1, students will be epected to: C47 eplore composition of functions C84 investigate and interpret the composition of functions Elaborations - Instructional Strategies/Suggestions (New material) Student groups should read and discuss p If outputs of a function g can be used as inputs of function f, then the result is a composition f(g()) {read as f of g of }. The stand alone notation is f B g which is read as f of g. Thus the value of f B g at is (f B g)() = f(g()). Definition If f and g are two functions, the composition of f and g,written f B g is defined by the equation (f B g)() = f(g()). There are a number of ways to determine the domain of the composite function: a) The domain of f must include the range of g. b) The domain of f B g is the set of all numbers in the domain of g such that g() ( the range of g ) is in the domain of f. For composition of functions defined by sets: Eample: Let g = {(1,4),(,5),(3,6)} and f = {(3,8),(4,9),(5,10)}. Find f B g. This can also be read as g(1) = 4, g() = 5, and g(3) = 6 so that only 4 and 5 in the range of g appears in the domain of f. So we get the following (f B g)(1) = 9 and (f B g)() = 10 Thus f B g = {(1,9),(,10)} For compositions of functions defined by equations: The domain of the composite function has the following restrictions: 1) any restrictions on the domain of g and ) any restrictions on f B g (you must first find the equation for f B g ) 1 4 f( ) = g( ) = + 1 Eample: Given and, find the composite function f B g and its domain. Step 1: The domain of g is { * 1} Thus 1 is not in the domain of f B g 1 1 Step : f g = = Thus! 1 is not in the domain of f B g. Therefore the domain of f B g is { 1,1} 114

65 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Composition of Functions (5.1) Group Activity For f ( ) =, g( ) = 1 a) g( + 1) b) g( + h) c) f(a + 3) find. Composition of Functions (5.1) Compositions of Functions Worksheet (5.1) Pencil/Paper For f() = + 4, g() =! find: a) f(g(0)) b) g(f(0)) c) g(g(!)) d) f(g()) e) g(f()) f) f(f()) Technology Graph f B g and g B f using the TI - 83 and make a conjecture about the domain of each of the following functions. Confirm your conjectures by finding formulas for f B g and g B f. a) f() = 5, g() = + 1 b) c) d) f g () =, () = 1 f g f( ) =, g( ) = ( ) =, ( ) =

66 Compositions of Functions Worksheet (5.1) p.74 Calculus tet 1. Use the table to evaluate each epression, if possible. If it is not possible, eplain why. a) g( f( )) b) ( f g)( 4) c) f( g( 3)) d) ( g f)( 6 ) e) ( g f)( 1) f) f( g( 0)). a) State the domain and range of f and g. Use the graphs of f and g to evaluate each epression, or eplain why the epression is undefined or cannot be evaluated. b) i) ( f g)( 1) ii) g( f( 1)) iii) f( g( 0)) iv) g( f( 0 )) v) ( g f)( ) vi) ( g f)( 5) 116

67 3. If f( ) = + 1 and g( ) = 3, find each of the following. a) f( 1) b) g( 9) c) f( 8r 6) d) f( g( )) e) gg ( ( )) f) ( f g)( ) g) ( g f)( ) h) i) g( f( 3 )) f( f( 0)) 4. If f( ) = 1 and g ( ) = 5, find each of the following, if it eists. F I a) g( 10) b) fhg 3 K J F HG F I HG K JI c) f g 5 7 KJ d) ( f g)( ) e) ( g f)( + 6) f) gg ( ( 0)) g) f( f( )) h) f( g( )) F HG F I HG K JI KJ i) g( f( )) j) f g 5 5. Find epressions for f g and g f for each pair of functions and state their domain. 1 a) f( ) =, g ( ) = 4+ 3 b) f( ) = 3, g( ) = + 6 c) f( ) =, g( ) = 5 d) f( ) = + 8, g( ) = e) f( ) =, g( ) = + 3 f) f( ) =, g( ) = + 5 g) f( ) =, g( ) = 3 h) f( ) = + 49, g( ) = 4 117

68 7. Given f( ) = and g( ) = + 4, find a) f( g( )) b) the domain of f( g( )) c) the range of f( g( )) d) g( f( )) e) the domain of g( f( )) f) the range of g( f( )) Etra Problems 1. For f() = + 5, g() =! 3 find: a) f(g(0)) b) g(f(0)) c) g(g(!)) d) f(g()) e) g(f()) f) f(f()). Graph f B g and g B f and make a conjecture about the domain of each of the following functions. Confirm your conjectures by finding formulas for f B g and g B f. a) f ( ) = 7, g( ) = b) f ( ) = 1, g( ) = c) f( ) = 3, g( ) = d) f( ) =, g( ) = Let g = {(1,4),(,5),(3,6)} and f = {(3,8),(4,9),(5,10)}. Find f B g. 4. Let f = {(!3,1),(0,4),(,0)}, g = {(!3,),(1,),(,6),(4,0)} and h = {(,4),(1,0)}. Find: a) f B g b) g B f c) f B h d) h B f + 1 f( ) = 3 1, g( ) = + 1 and h( ) = 5. Let. Find and simplify: a) f(g(!1)) b) (h B f)(!7) c) (h B g B f)() 3 f ( ) =, g( ) = 1 and h( ) = 6. Let. Find each of the composition functions below 118

69 and state their domain: a) f B g b) g B f c) h B g 7. Let f() =, g() = 3 and h() =. Write each function below as a composition of the above three functions. a) F( ) = 3 b) G ( ) = 6 c) H( ) = 3 8. Let f() =,g() = 7and h() =. Write each of the following functions as a composition of functions f, g and h. a) F( ) = 7 b) G ( ) = 7 c) d) e) H ( ) = 7 M ( ) = P ( ) = ( 7) f) T( ) = 7 g) L ( ) = 7 7 h) C ( ) = ( 7) 9. Determine whether the following are true or false: a) If f(3) = 19 and g(19) = 99, then (g B f)(3) = 99. b) If f() = and g() =,then (f g)() =. c) If f() = 5 and g() = /5, then (f B g) () = (g B f)() =. d) If n() = (!9), s() = and h() =!9, then n = h B s. e) If f() = and g() =, then the domain of f B g is [,4). 119

70 The function g() = converts from shoe sizes in Canada to shoe sizes in France g(). The function f() = converts from shoe sizes in France to shoe sizes in Japan f(). Find the function h() that will convert shoes sizes in Canada to shoe sizes in Japan h(). Use this function to complete the following table. 11. The function g(d) = 0d converts Canadian dollars into Greek drachmas while the function l(g) = 6g converts Greek drachmas to Italian lira. Find the composite function that will convert Canadian dollars directly to Italian lira. How many lira will $500 Canadian dollars buy? How many Canadian dollars will 50,000 lira buy? Use this function to complete the table below. 10

71 1. a) f( g( 0)) = b) g( f( 0)) = c) gg ( ( )) = 6 d) f( g( )) = + e) gf ( ( )) = f) f( f( )) = + 10 Answers for Compositions of Functions Worksheet (5.1) - Etra problems 11

72 . a) D = [0, 4) f B g = [7, 4) g B f D D 1

73 b) D = [0, 4) f B g D = [0, 4) g B f 13

74 c) D = [!, 4) f B g D = (!4,!1) c (1, 4) g B f 14

75 d) D = (!4,4), f B g D = (!4,4),!3 g B f B g = {(1,9),(,10)} 3. f 15

76 4. a) f B g = {(!3,0),(1,0),(4,4)} b) g B f = {(!3,),(0,0)} c) f B h = {(1,4)} d) h B f = {(!3,0)} 5. a) (f B g)(!1) = 5 b) (h B f)(!7) =!7 c) (h B g B f)() = 3! a) f g = 1 1 [, ) b) g f = 1 [ 0, ) c) h g = ( 1) (, ) 7. a) f B g b) g B g c) g B h B f 8. a) g B h b) g B f c) f B g B h d) h B g e) h B g B f f) f B g g) g B f B g h) h B g B h 9. a) T b) F c) T d) F e) T 16

77 10. h() = (f B g)() = (l B g)() = 130d 17

78 SCO: By the end of Grade 1, students will be epected to: C96 eamine, interpret and apply the relationship between functions and their inverses Elaborations - Instructional Strategies/Suggestions (New material) It is possible for one function to undo what another function does. For eample, squaring undoes the operation of taking a square root. The composition of two such functions is the identity function. If a function has two ordered pairs with different first coordinates and the same second coordinate, then the function is not one-to-one(1-1). Therefore it has no inverse function. The graph of a 1-1 function never has the same y-coordinate for two different -coordinates. The inverse of a 1-1 function f is the function f!1 (read f inverse ), where the ordered pairs of f!1 are obtained by interchanging the coordinates in each ordered pair of f. Geometrically this means that the graph of a 1-1 function and its inverse are reflections of each other through the identity function(y = line); see Math 61B guide Unit1 p.1). (See Identity Function Sheet at end of unit). Horizontal Line Test If each horizontal line crosses the graph of a function at no more than one point, then the function is a 1-1 function and its inverse is a function also. If a horizontal line crosses the graph of a function at more than one point then the function is not a 1-1 function and its inverse is not a function but simply a relation. E. f() = f() = 3 not a 1-1 function thus its inverse is a relation and not a function this is a 1-1 function and its inverse is a function Definition of inverse A function g is the inverse of a 1-1 function f iff:: a) the domain of g = the range of f b) (g B f )() = for any in the domain of f. 18

79 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Inverses of Functions Group Discussion True of False? Eplain. 1. The inverse of the function {(,3),(5,5)} is {(5,),(5,3)}.. The function f() =! is a 1-1 (invertible) function. Inverses of Functions Inverses of Functions Worksheet Inverse Function Theory 3. The inverse of a function is always an inverse. 4. The function f() = 4 is 1-1(invertible). y = 5. According to the horizontal line test, is 1-1. Pencil/Paper Is f = {(1,3),(,4),(5,7)}a 1-1 function? Find f!1, f!1 (3), and (f!1 B f )(1). Pencil/Paper Determine whether the function f() = + 5 is 1-1. If it is 1-1, find its inverse function. Group Activity Find (g B f)() and determine whether g is the inverse of f for 3 3 f ( ) = and g( ) =. Pencil/Paper Use the horizontal line test to determine whether each function is 1-1. a) b) 19

80 1. Read pages on inverse function theory. Inverses of Functions Worksheet True of False? Eplain.. The inverse of the function {(,3),(5,5)} is {(5,),(5,3)}. 3. The function f() =! is a 1-1 (invertible) function. 4. The inverse of a function is always a function. 5. The function f() = 4 is 1-1(invertible). y = 6. According to the horizontal line test, is Determine whether each function is 1-1. If it is 1-1, find its inverse function. a) {(!1,0),(0,0),(1,0)} b) {(3,0),(,5),(4,6),(7,9)} c) {(1,1),(,),(4.5,4.5)} d) {(!,3),(4,5),(,3)} e) {(3,1),(5,),(7,4),(9,8)} 8. Is f = {(1,3),(,4),(5,7)}a 1-1 function. Find f!1, f!1 (3), and (f!1 B f )(1). 9. Use the horizontal line test to determine whether each function is 1-1. a) b) c) d) e) f) 10. Determine whether each function is 1-1. If it is 1-1, find its inverse function. a) f() = + 5 b) f() = c) f() = 5 d) f() = 3 7 e) f() =

81 f) f() = + 1 g) f() = 3 1 h) f() = 7 1 i) f() = 11. Each of the following functions is 1-1. Describe the inverse in words. a) f() = 4 b) f() =! 19 c) f() = 3! 8 d) f() = ½! 6 1. Find (g B f)() and determine whether g is the inverse of f. a) f() = and g() =. b) f() = 3+ 4,g() = 1 c) f() = 0 5,g() = d) e) f) f() = + 1,g() = f() = + 3,g() = f() = 7,g() = Find the inverse of each of the following functions and graph both f and f!1 on the same coordinate aes. Use the nd Draw 8:DrawInv feature on the TI-83 to verify your results. Use a Reflect-View to verify the results as well. Is the inverse a function? a) f() = 3+ b) c) f() = 4 for 0 f() = 1 for 0 d) f() = e) f() =

82 . F 3. F 4. F 5. F 6. F 7. a) no b) yes f 1 = c) yes f 1 = d) no e) yes f 1 = {( 03, ), ( 5, ), ( 64, ), ( 97, )} {( 11, ), (, ), ( 4545.,. )} {( 13, ),( 5, ),( 47, ),( 89, )} Answers for Inverses of Functions Worksheet 8. yes f 1 = {( 31,),( 4, ),( 75, )} f 1 1 ( 3) = 1 ( f f)( 1) = 1 9. a) no b) no c) yes d) yes e) no f) yes 10. a) yes 1 5 f ( ) = b) no c) yes 1 f ( ) = 5 d) yes f ( ) = 3 e) yes 1 3 f ( ) = f) yes f 1 ( ) = g) yes f ( ) = h) yes f ( ) = 7 13

83 i) yes f ( ) = Requirements for Inverse Functions j) yes f + ( ) = 1 1. function is 1-1 k) yes f ( ) =. D g = R f 3. (g B f)() = a) f ( ) = 4 1 b) f ( ) = + 19 c) f ( ) = d) f ( ) = ( + 6) 1. a) ( g f)( ) = Req. 1 met, Req. 3 met, D g R f thus g is not an inverse of f b) ( g f)( ) = Req. 1 met, Req. met, Req. 3 not met c) ( g f)( ) = All Req. met, thus g is the inverse function of f d) ( g f)( ) = Req. 1 not met, Req. not met, Req. 3 met, thus not an inverse function e) ( g f)( ) = Req. 1 met, Req. met, Req. 3 met, thus is an inverse function f) ( g f)( ) = Req.1 met, Req. met, Req. 3 not met, thus not inverse function a) f ( ) = 3 inverse function b) f 1 ( ) = + 4 inverse function c) f 1 ( ) = 1 not an inverse function d) f 1 ( ) = 3 inverse function e) f ( ) = not an inverse function 133

84 Inverse Function Theory Many actions in the real world are reversible, a closed door can be opened, an open window can be closed or a light can be turned on. However, not all actions are reversible, for eample a hurricane or an eplosion or saying something nasty to a friend. In mathematics this basic concept of reversing operations to arrive back at some original epression is associated with the term inverse. In communications a mathematical function is used to send voices and pictures over long distances. An inverse function must decode the signal back into sounds and pictures that are clear and static free. The first manned landing on the moon sent pictures back to Earth that were very poor in quality. Scientists set to work to rectify( no pun intended) the problem. They discovered that the Fourier Transform function allows for over 90% of a signal to be lost without appreciatively affecting picture or sound quality. The second mission to the moon sent back crystal-clear sound and pictures using this new discovery. Inverse operations are important to understanding inverse functions. Many government agencies have used codes to send sensitive information from one place to another for thousands of years. Even in the days of the Roman Empire codes were used to good effect. Every coding system requires a consistent set of rules to code and decode messages. These rules are guarded under high security by governments and industry alike. Spies abound, not just looking for state secrets but for technological secrets in industry as well. Coding and decoding are inverse actions or operations. It is important that any code system yield only one output for each input. They must be 1-1 functions, something we will discuss in more detail later. Actions and their inverses occur in everyday life. A person opens the car door, gets in, and starts the engine. The inverse actions are: turn off the car, get out of the car and shuts the car door. Notice that not only are the inverse actions carried out but they are done in the reverse order. In math there are basic operations that are inverses of each other: < addition and subtraction < multiplying and dividing < squaring and square rooting An understanding of inverse operations is key to solving equations. Reading the left side of the equation below; E = 11 tells us that has been: 1. multiplied by. 3 added to the result from 1 therefore to solve the original equation for we must: 1. subtract 3. divide by 134

85 To find the inverse of a function defined by ordered pairs we simply interchange the and y coordinates. E. f = {(1,),(3,5),(4,7)} the inverse is f 1 = {( 1, ), ( 53, ), ( 74, )} The original function must have each output (y-value) assigned to only one input (-value). These types of functions are called 1-1 functions and their inverses can be called inverse functions. Eample above is a 1-1 function. The function f() = yields an output of 4 for an input of either + or! and is thus not a 1-1 function. It can have an inverse but that inverse will not be a function. The Horizontal Line Test is used to test whether or not a function is 1-1 if the function s graph is given or can be visualized. Geometrically, this amounts to reflecting the ordered pairs in f across the y = line (see Math 61B guide, Unit 1 p.1). Any point on a 1-1 function and its corresponding point on the inverse function are equal perpendicular distances from the y = line. Hence a Reflect - View would show these points as mirror images of each other across the y = line. If we need to find the inverse of a function in symbolic form then we may be able to find it: < mentally < using a step - by - step procedure < graphically 1. Mentally - By understanding what operations have been done to the variable in the function and undoing them in the reverse order we may obtain the inverse of the function. If the original function is 1-1 then the inverse will be a function as well. E. 3 f() = + 3 operations done to are: 1. multiply by. add 3 If we subtract 3 from and then divide by we will get its inverse. 1 3 f ( ) = Note: The f -!1 label can be used for this inverse because the original function is a 1-1 function.. using a step - by - step procedure - The steps are as follows; < determine whether or not the function is 1-1 < replace f() with y < interchange and y < solve for y < replace y with f!1 () < check that the domain of the inverse equals the range of the original function: D 1 = R or D = R < ( g f)( ) = f f g f 135

86 If the function in E 3 were graphed, we could see that it is in fact 1-1. The graph of the function and its inverse are equal perpendicular distances from the identity function ( the y = line). When 1-1 functions are composed with their inverse functions, f B g or g B f, the result is that the composition yields the identity function f() =. 3. Graphically - not all functions written in symbolic form lend themselves to the above step - by - step procedure. For eample f() = ! + 5! 3 is a 1-1 function but it would be impossible to obtain a symbolic representation for its inverse f!1. Graphically we can find its inverse using a Reflect- View. An eample is shown above that was done with the graphing calculator but could also have been done using the Reflect-View. 136

87 137

88 SCO: By the end of Grade 1, students will be epected to: C eplore and apply functional relationships and notation, both formally and informally Elaborations - Instructional Strategies/Suggestions Etending the Function Toolkit Challenge student groups to use their knowledge of domain and range, written in interval notation, to determine the natural domain and range of the following classes of functions: Power Functions: < polynomial function! constant function! identity function (see note at unit end)! linear function! quadratic function... < radical function < rational function Piecewise Functions: < absolute value function < step function < general function Transcendental Functions(non-algebraic functions) < trigonometric function < logarithmic function < eponential function A general piecewise function is a function that requires more than one equation to define it. The function behaves differently on some intervals from the way it behaves on others. A step function is a piecewise function which is constant on each of the intervals for which it has a special definition. Student groups should investigate the Etending the Functions Worksheet. 138

89 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Etending the Function Toolkit Group Discussion Sketch the graph for each function, state the natural domain and determine whether or not each function is 1-1. a) f( ) = b) f( ) = + 6 c) f( ) = Etending the Function Toolkit Etending the Function Toolkit Worksheet Presentation Match each graph to the function list. a) constant function b) quadratic function c) piecewise function d) eponential function e) rational function f) identity function a) b) c) d) 139

90 Etending the Function Toolkit Worksheet 1. Sketch the graph for each function, state the natural domain and determine whether or not each function is 1-1. a) f( ) = + 3 b) f( ) = c) f( ) = d) f( ) = e) f( ) = 3 f) f( ) = g) f( ) = h) f( ) = i) f( ) = 1 j) f( ) = k) f( ) = 1 3 l) f( ) = 4 1 m) f( ) = n) f( ) = 3 + o) f( ) = R S T + 1 [ 11, ] = 1 (, 1 ) 140

91 . Match each graph to the function list. a) constant function f) identity function k) trigonometric function b) quadratic function g) piecewise function l) quartic function c) radical function h) rational function m) cubic function d) absolute value function i) eponential function e) linear function j) logarithmic function a) b) c) d) e) f) g) h) i) j) k) l) m) 141

92 Answers 1. a) D = (, ) 1 1 b) D = (, ) not 1 1 c) D = (, ) not 1 1 d) D = (, ) not 1 1 e) D = [ 3, ) 1 1 f) D = [ 1, ) 1 1 g) D = (, 3) ( 3, ) 1 1 h) D = (, 1) ( 1, ) 1 1 i) D = (, ), 0 not 1 1 j) D = (, ), 3, not 1 1 k) D = (, 3) 1 1 l) D = (, ) not 1 1 m) D = (, ) not 1 1 n) D = (, ) not 1 1 o) D = [ 1, ) not 1 1. a) graph (b) b) graph (d) c) graph (e) d) graph (a) e) graph (j) f) graph (m) g) graph (f) h) graph (c) i) graph (g) j) graph (l) k) graph (i) l) graph (k) m) graph (h) 14

93 Identity Function One of the most basic functions is the identity function. This function has ordered pairs with identical and y values. It is defined by y = or f() =. This function will be discussed in a later section on Inverses of Functions. It is the basis upon which many other functions are built. The section on Operations with Functions (3.7) will work with this and other functions. Eamples are shown below. f ( ) = + 3 f ( ) = 4 is the sum of the identity function and the constant function 3. is the product of the identity function and the constant function 4. f ( ) = is the square of the identity function. 143

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