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Date Created: 10/11/15
Math 242 Enrichment Problems Section 14 Functions As Mappings We have de ned a function as simply a relation that meets the condition that each domain value is paired with one and only one range value We can also think of a function as a mapping between two sets A and B We write f A gt B which is read fmaps A into B SetA is the domain of the function and set B is called the codomain If x y is an ordered pair belonging to the function then we write fxy where y is called the image of x under f The range is the set of all image values SetB the codomain may be the range itself or a set that contains the range If B is the range then we say that f maps A onto EXAMPLE A Function as aMappz39ng The function f x x2 is a mapping from the real numbers into the real numbers That is each domain value and each range value will be a real number We can write f R gt R Since f 3 9 9 is the image of 3 under f Although every real number is an element of the domain not every real number is included in the range Because the square of a real number is nonnegative the range of f is the interval 0 00 In other words 0 00 is the image of R under f We can say more specifically that f is a mapping of R onto 0 00 and write f R gt 0 oo EXERISES 1 Given f x x2 4x l find the image of each value or expression under f Simplify completely a 3 b 21 c 6 d r l e x h 2 Given f x x2 explain the difference between the statement f R gt R and f R gt 0 00 Find the image of R under each function Write each function as a mapping from R onto its range 3 f x x3 4 hx x4 5 gmhyz Math 242 1 Group Problems Linear Functions and Variation After 8000 miles of use a set of brake pads for an automobile were 75 millimeters thick At 20000 miles the pads were down to 375 millimeters Assume the relationship between the pad thickness and the number of miles driven is linear Let P be the pad thickness in mm and x the number of miles driven in thousands Write P as a function of x b At what rate are the pads wearing down c How thick were the pads when new d If the pads should be replaced at 2 mm when should the owner take the vehicle in for service e The owner decides not to replace the pads at 2 mm How many more miles can be driven until the pads are completely worn down f What is the practical domain for this function g Write your equation in a as a direct variation Translate this variation into words A s V Newton s Law of Cooling states that the rate r at which the temperature of an object decreases varies directly with the difference between the temperature of the object T and the temperature of the surrounding environment 70 F From class lecture we have rT kT 70 kT 70k a Sketch a graph of r against T use Tfor the horizontal axis Assume the constant of variation is positive Label both intercepts b What is the signi cance of the horizontal intercept Math 242 Group Problems Section 82 A stochastic matrix also called a transition matrix is a matrix whose entries represent the probability that a transition from one state to another state will occur All of the entries in a stochastic matrix must be real numbers between 0 and l inclusive That is 0 S 61 S l for all values of 139 and j In addition all of the entries in each column must add to 1 Consider the matrix P which is a stochastic matrix Each column represents percentages of voters registered with a particular political party Each row represents percentages of voters that will align themselves with the indicated party during the next election The matrix V gives the initial number of registered voters for each party Currently R D I 080 012 025 Republican NXt 5500 R e P Democratic election V D 005 002 060 Independent 400 1 For example the rst column in P tells us that 80 of registered republicans will stay loyal to their party in the next election but 15 of republicans will switch to the Democratic Party and the remaining 5 will switch to an independent party What percentage of registered democrats will switch to the Republican Party in the next election What percentage of independents will stay loyal to their party in the next election How many voters are initially democrats What percentage of voters are initially republicans Find the product PV by hand write it out What do the entries in PV represent 959 How many registered voters are republicans after the rst election What percentage of voters are republicans after the rst election 6 Use your calculator to nd PZV Copy it down What does the rst entry in PZV represent 7 Use your calculator to nd P2 Copy it down What does each entry in the rst column represent Use your calculator to nd P30 and P31 Notice that the entries in these two matrices are essentially the same With each additional election the voting preferences will not change substantially We say that the population has achieved a steady state Also notice that the values in each row are nearly identical 8 In the long run what percentage of registered voters will belong to each of the three parties 2008 Turner Educational Publishing Math 242 GRAPES 0F BASIC ALGEBRAIC FUNCTIONS c mun Functh xanan Funcum Quuuuuc Functh m 0 m x xx1 Dammn m m Dammn m m Range m 7 mug mu 0th Fuucuuu sum FmFuucuuu Cube mean m x x fwd7 Dammr 7m m Range m m Dumun n u Range Fug Dammn u u Range 7 w Recxpmcal Functh m m ouan 4M mug W gm SquareRecxpmcal Functh fxlx1 Dammn 4M Fang w AbsaluzeValuel mcum m x Dammn 7 Range n Sa39mcn39deFuncum 4L Dmmn 711 Raw m H Math 242 Enrichment Problems Section 93 Iteration and Geometric Sequences If we know any term in a geometric sequence we can easily nd the next term by multiplying by the common ratio r We can use this property of a geometric sequence to de ne the sequence recursively GEOMETRIC SEQUENCE RECURSIVE FORMULA A geometric sequence an with rst term a and common ratio r can be de ned recursively as 611 61 am ran rt 21 Notice that the recursive formula an1 ran describes a linear relationship or direct variation between consecutive terms which we can interpret in the context of iteration If we de ne fx rx then the geometric sequence an is the orbit of the seed x0 a1 underf The graph offis a line passing through the origin with a slope of r Figure 1 shows a cobweb diagram of the orbit Because of the slope the points either get closer to one another or spread further apart Every geometric sequence can be interpreted as the orbit of some seed under a linear function passing through the origin with slope of r y fx rx a3 a4 Hr 02 a3 01 a2 A a1 a2 a3 614 Figure 1 EXERISES 1 Consider the geometric sequence an 4 2 l from the class lecture a This sequence is the orbit of what seed under what function b Sketch a cobweb diagram for the rst five terms of the sequence 2 Consider the geometric sequence an 3 l2 48 l92 768 from the class lecture a This sequence is the orbit of what seed under what function b Sketch a cobweb diagram for the first three terms of the sequence 3 Let gx bx for b gt 0 and b 7 1 Since g is an exponential function if we restrict the domain of g to the natural numbers we will obtain a geometric sequence a Write the rst ve terms of the geometric sequence generated by g b This sequence is the orbit of what seed under what function c Sketch a representative cobweb diagram for the rst four terms of the sequence if b lt l d Sketch a representative cobweb diagram for the rst four terms of the sequence if b gt 1 e Use your graphs from c and d to explain why exponential growth results when b gt 1 but exponential decay results when b lt l 2008 Turner Educational Publishing
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