FUNCTIONS WORKSHEET Name:MA 202Spring Semester 2004WARNING: You must SHOW ALL OF YOUR WORK. You will receive NO CREDITif you do not show your work.DUE: Thursday, 05 February 20031. Jake is is y years old. Write an algebraic expression with the variable y for Joni’s ageif:(a) Joni is three years older than Jake was five years ago.(b) Joni is twice as old as Jake will be in six years.(c) Joni is half as old as Jake’s mother who is three times as old as Jake was fouryears ago.2. Consider the four number machines shown below.Machine A takes a number x and returns x + 3. For example, if we place the number5 in machine A, it will return the number 8. If we place the number −1.2 in machineA, it will return the number 1.8. Machines B, C, and D act similarly. If we placethe number 4 in machine B, it will return the number . If we place thenumber −7 in machine C, it will return the number . If we place thenumber 5.2 in machine D, it will return the number .Machines A, B, C, and D are simple machines because they only perform one operationon a number. Let’s take a look at a machine which is not a simple machine. Considermachine E which is shown below.1Machine E is a complex machine because it performs more than one operation on thenumber that is input. If we place the number x into machine E, it will first add three tox and then square (x + 3) yielding (x + 3)2. We can construct E from simple machines.For example, we can input a number into machine A and then place the output frommachine A into machine B. When machines A and B are linked in this way, togetherthey perform the same operations that machine E performs.I have described several complex machines below. You need to construct each complexmachine from the simple machines A, B, C, and D. Your answer should be a diagramsimilar to the ones shown above.• When the number x is placed in machine F , it returns(x + 3)22.2• When the number x is placed in machine G, it returnsx2+ 32.• When the number x is placed in machine H, it returns 5x + 3.• When the number x is placed in machine I, it returns 5(x + 3).• When the number x is placed in machine J, it returns(5x + 3)22.Mathematicians are, by nature, somewhat lazy when it comes to writing. We reallylike shorthand — provided that it is used correctly. (Remember that you need to bevery, very careful with the = symbol. You may only use it to mean that quantitiesare, in fact, equal. You may not use this symbol for the word “implies.”) It requiresa lot of time to draw number machines, especially if you are using a computer as Ihave been doing to write this worksheet. The shorthand for number machines is calledfunction notation. Here is how it works. Consider number machine A. The name ofthis machine is “A.” It takes the input x and returns the value x + 3. Therefore theshorthand for machine A is A(x) = x + 3. Since A(x) = x + 3, we can either say thatthe output of machine A is x + 3 or A(x). If we want to put a specific number into themachine we can replace the input variable x with our number. For example, if we putthe number 5 into machine A, it will return the value 8. We represent this by A(5) = 8.Notice that A(5) = 5 + 3, that is 5 replaces all of the x values in our notation.• Write the function notation for machines B–J.• What is J(2)?• What is C(−5)?• What is D(4)?• What is I(x2+ h)?• What is B(x + 3)?• What is B(A(x))?Recall that we can construct machines E–J from our simple machines. How would werepresent this using function notation? Let’s take a look at machine E. We constructedE from simple machines by joining machine A to machine B. We input the value x intomachine A and it returns the value A(x). Then this value goes into machine B, thatis A(x) goes into machine B. Therefore B should return the value B(A(x)). Hence, wecan write E(x) = B(A(x)). This function notation is used to show how the complexmachine E can be constructed from the simple machines A and B.• Each of the complex machines F–J can be constructed from the simple machinesA–D. Describe these constructions using function notation.Now let’s see how we can use these machines to help us solve equations. Suppose wewant to solve(x + 3)2= 5.3Notice that all the x’s are on one side of the equation. Also notice that the left hand sideof the equation is E(x). To solve this equation, we need to walk backwards through thesimple machine construction. Remember, that we constructed E from simple machinesby sending a number through machine A and then through B. So if we are going towalk backwards through this construction we will need to undo the operation in B andthen undo the operation in A. Since B squares a number, we will need to undo thisoperation by taking the square root of both sides of the equation. Note that we willhave to be careful here to include both the positive and negative square root. Thisgives:(x + 3)2= 5p(x + 3)2=√5x + 3 = ±√5Now we need to undo the operation of machine A. Since A adds three to a number, wewill need to undo this operation by subtracting three from both sides of the equation.This gives:(x + 3)2= 5p(x + 3)2=√5x + 3 = ±√5x + 3 − 3 = ±√5 − 3x = ±√5 − 3• Verify that√5 − 3 and −√5 − 3 are both solutions to the equation (x + 3)2= 5.We can use this process of walking backwards through simple machines to solve lotsof equations. Solve the following equations. Explain to your group members how yoursolutions are related to walking backwards through simple machines.• 16 =(x + 3)22.•12=x2+ 32.• x + 4.2 = 6x + 3.• 5(x + 3) = 7.•(5x + 3)22=13.3. For each complex function described below, describe the simple machines that areneeded to construct the complex function. Then construct the complex machine fromthe simple machines you described.4(a) k(x) =2x+35(b) l(x) =1x+2(c) m(x) =√x2− 1(d) n(x) = 3√4x − 5 − 74. Find the solution set for each equation. Be sure to show all of your work and checkyour answers.(a) 4x + 6 = 9x − 2(b)2x+35=110(c)1x+2= 7(d)1x+2= 0(e)√x2− 1 = 0(f) 3√4x − 5 − 7 = −15. Which of the following equations are identities? Which are conditional equations?Which equations do not have real number solutions?(a) x + 2 = 53(b) 2x + x2= x(2 + x)(c) (x + y)2= x2+ y2(Be careful here.)(d) x2=11x2(e) x2+ 1 = 0.(f)√x2− 1 = 06. Consider the sequence of polygon trains shown below. Write an algebraic expressionusing the variable n to represent the number of edges in the nthpolygon