SECTION 2-6 Inverse Functions
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1. FIGURE 9 y y A A 5 5 5 gt X ZE 3 gt x 4x x f x 4x x2 x lt 2 a b Step 2 Solve the equation y f x for x y 4x x x 4x y Rearrange terms x 4x 4 y 4 Add 4 to complete the square on the left side x 2 4 y Taking the square root of both sides of this last equation we obtain two pos sible solutions x 2 V4 y The restricted domain of f tells us which solution to use Since x 2 implies x 2 0 we must choose the negative square root Thus 194 2 Graphs and Functions Matched Problem 3 x 2 V4 y x 2 V4 y and we have found t QS VA Hy Step 3 Interchange x and y y f 2 V4 x Step 4 Find the domain of f The equation f x 2 V4 x is defined for x lt 4 From the graph in Figure 9 b the range of f also is 4 Thus fl 2 V4 x x4 The check is left for the reader The graphs of f f and y x are shown in Figure 10 Sometimes it is diffi cult to visualize the reflection of the graph of f in the line y x Choosing some points on the graph of f and plotting their reflections first makes it easier to sketch the graph of f Figure 11 shows a check on a graphing utility 5 76 7 6 FIGURE 10 FIGURE 11 Find the inverse of f x 4x x x 2 Graph f f and y x in the same coor2. y f x y x f x 5 2 N 5y 5 2 5 10 a b and b a f x 2x 1 f x vVx 1 are symmetric with f x 4x44 f x x 1 x 0 respect to the line y x a b c 192 2 Graphs and Functions Theorem 6 FIGURE 8 Restricting the domain of a function Symmetry Property for the Graphs of f and f The graphs of y f x and y f x are symmetric with respect to the line y x Knowledge of this symmetry property makes it easy to graph f if the graph of f is known and vice versa Figures 7 b and 7 c illustrate this property for the two inverse functions we found earlier in this section If a function is not one to one we usually can restrict the domain of the func tion to produce a new function that is one to one Then we can find an inverse for the restricted function Suppose we start with f x x 4 Since f is not one to one f does not exist Fig 8 a But there are many ways the domain of f can be restricted to obtain a one to one function Figures 8 b and 8 c illustrate two such restrictions f x x 4 g x x 4 x 0 h x x 4 x lt 0 f t does not exist gx Vx 4 x 4 h 4 x Vx 4 x 4 a b c EXPLORE DISCUSS 4 To grap
3. y x x for x 0 Find f for f x Vx 2 EXPLORE DISCUSS 3 Most basic arithmetic operations can be reversed by performing a second opera tion subtraction reverses addition division reverses multiplication squaring reverses taking the square root etc Viewing a function as a sequence of reversible operations gives additional insight into the inverse function concept For example the function f x 2x 1 can be described verbally as a function that multiplies each domain element by 2 and then subtracts 1 Reversing this sequence describes a function g that adds 1 to each domain element and then divides by 2 or g x x 1 2 which is the inverse of the function f For each of the follow ing functions write a verbal description of the function reverse your description and write the resulting algebraic equation Verify that the result is the inverse of the original function 1 A f 3x 5 B f vx 1 f re FIGURE 7 Symmetry with respect to the line y x There is an important relationship between the graph of any function and its inverse that is based on the following observation In a rectangular coordinate sys tem the points a b and b a are symmetric with respect to the line y x see Fig 7 a Theorem 6 is an immediate consequence of this observation
4. 4 59 f y 2 V3 x 60 f 4 Vet5 61 How are the x and y intercepts of a function and its inverse related 62 Does a constant function have an inverse Explain C The functions in Problems 63 66 are one to one Find f 63 fx x 1 2 x 1 64 fy 3 x 5F x55 65 f y 2x 2 x s 1 66 f x x 8x 7 x 4 The graph of each function in Problems 67 70 is one quar ter of the graph of the circle with radius 3 and center 0 0 Find f find the domain and range of f and sketch the graphs of f and f in the same coordinate system 67 fe VI Z 0 lt x53 68 fx V9 wX 0Sx53 2 6 Inverse Functions 197 69 fx V9 xX 3 5x50 70 f V9 x 3 5x50 The graph of each function in Problems 71 74 is one quar ter of the graph of the circle with radius 1 and center 0 1 Find f find the domain and range of f and sketch the graphs of f and f in the same coordinate system 71 fa 1 V1I 0Sx 1 72 fx 1 VI 0Sx 1 73 fM 1 V1 1sx lt 0 74 fa 14 V1 x 1 x 0 75 Find f x for f x ax b a 0 76 Find f x for f Va 7 a gt 0 0SxSa 77 Refer to Problem 75 For which a and b is fits own inverse 78 How could you recognize the graph of a function that is its own inverse 79 Show that the line through the points a b and b a a b is perpendicular to the line y x see the figure 80 Show that the point a
5. mum profit occur Do the maximum revenue and maximum profit occur for the same output Discuss
6. Domain Range Domain Range Domain Range 3 0 0 gt 3 0 ees 5 2 ere 2 gt 5 4 q n fis not a function g is a function but is not h is a one to one one to one function EXAMPLE 1 Solutions Determining Whether a Function Is One to One Determine whether f is a one to one function for A f f 2x 1 A To show that a function is not one to one all we have to do is find two differ ent ordered pairs in the function with the same second component and different first components Since fe and f 2 2 4 the ordered pairs 2 4 and 2 4 both belong to f and f is not one to one B To show that a function is one to one we have to show that no two ordered pairs have the same second component and different first components To do this we assume there are two ordered pairs a f a and b f b in f with the same sec ond components and then show that the first components must also be the same That is we show that f a f b implies a b We proceed as follows 184 2 Graphs and Functions f a f b Assume second components are equal 2a 1 2b 1 Evaluate f a and f b 2a 2b Simplify a b Conclusion f is one to one Thus by Definition 1 f is a one to one function Matched Problem 1 Determine whether f is a one to one function for A fQ 4 r B f x 4 2x The methods used in the solution of Example 1 can be stated as a theorem Theorem
7. inverse of a function f read f inverse is a special symbol used here to represent the inverse of the function f It does not mean 1 f 2 6 Inverse Functions 187 DEFINITION 2 Inverse of a Function If f is a one to one function then the inverse of f denoted f is the function formed by reversing all the ordered pairs in f Thus f 40 2 y is in f If f is not one to one then f does not have an inverse and f does not exist The following properties of inverse functions follow directly from the definition Theorem 4 Properties of Inverse Functions If f exists then 1 f is a one to one function 2 Domain of f Range of f 3 Range of f Domain of f EXPLORE DISCUSS 1 Most graphing utilities have a routine usually denoted by Draw Inverse or an abbreviation of this phrase consult your manual that will draw the graph formed by reversing the ordered pairs of all the points on the graph of a function For exam ple Figure 4 a shows the graph of f x 2x 1 along with the graph obtained by using the Draw Inverse routine Figure 4 b does the same for f x x A Is the graph produced by Draw Inverse in Figure 4 a the graph of a func tion Does f exist Explain B Is the graph produced by Draw Inverse in Figure 4 b the graph of a func tion Does f exist Explain If you have a graphing utility with a Draw Inverse routine apply it to the graphs of y
8. s manual for your particular graphing utility and discuss among the members of the This group project may be done without the use of a graphing utility but significant additional insight into mathematical modeling will be gained if one is available Chapter 2 Group Activity 199 group how this is done After obtaining the linear regression line for the data in Table 2 graph the line and the data in the same viewing window The linear regression line found in part 1 is a mathematical model for the price demand function and is given by P x 666 5 21 5x Price demand function Graph the data points from Table 2 and the price demand function in the same rectangular coordinate system The linear regression line defines a linear price demand function Interpret the slope of the function Dis cuss its domain and range Using the mathematical model determine the price for a demand of 10 000 bikes For a demand of 20 000 bikes B Building a Mathematical Model for Cost Plot the data in Table 3 in a rectangular coordinate system Which type of function appears to best fit the data 1 Fit a linear regression line to the data in Table 3 Then plot the data points and the line in the same view ing window The linear regression line found in part 1 is a mathematical model for the cost function and is given by C x 86x 1 782 Cost function Graph the data points from Table 3 and the cost function in the same rectangula
9. x Vxt 1 m Problems 23 30 require the use of a graphing utility Graph each function and use the graph to determine if the function is one to one 2 xX xl 23 f ma 28 fx I 2_4 27 fix i 18 G x ix 1 20 K x V4 x 22 Nx x 1 x xl 24 f x _ bP bl 26 fo In Problems 35 40 verify that g is the inverse of the 5 one to one function f by showing that g f x x and 28 f x i flg x x Sketch the graphs of f g and the line y x x 1 in the same coordinate system Check your graphs in Problems 35 40 by graphing f g and the line y x in a squared viewing window on a graphing utility 35 f x 3x 6 g x 5x 2 36 f x x 2 g x 2x 4 37 f 4 x x 0 9 Vx 4 38 f x Vx 2 a x x 2 x20 39 f Vx 239 P 2 x 50 40 f 6 xL x lt 0 g x V6 x The functions in Problems 41 60 are one to one Find f 41 f x 4x 42 f x 4x 43 f x 2x 7 44 f x 0 25x 2 25 45 f x 0 2x 0 4 46 f x 7 8x 47 A 3S2 48 5 E x x 2x 4x 49 f x 50 f x 5 51 f amp 0 2x 0 4 52 f x 02 0 lx 0 5 x 0 5 53 f x 8 5 54 f x 20 9 55 fx 2 W3x 7 56 fix 1 W4 5x 57 f x 2V9 x 58 fix 3Vx
10. 1 One to One Functions 1 If f a f b for at least one pair of domain values a and b a b then f is not one to one 2 If the assumption f a f b always implies that the domain values a and b are equal then f is one to one Applying Theorem 1 is not always easy try testing f x x 2x 3 for example However if we are given the graph of a function then there is a simple graphic procedure for determining if the function is one to one If a horizontal line intersects the graph of a function in more than one point then the function is not one to one as shown in Figure 1 a However if each horizontal line intersects the graph in one point or not at all then the function is one to one as shown in Figure 1 b These observations form the basis for the horizontal line test FIGURE 1 Intersections of graphs y y and horizontal lines A A y f x a f a b f b a f a l l l i 1 F x f x x a b a f a f b fora b Only one point has ordinate f isnot one to one f a f is one to one a b 2 6 Inverse Functions 185 Theorem 2 Horizontal Line Test A function is one to one if and only if each horizontal line intersects the graph of the function in at most one point The graphs of the functions considered in Example 1 are shown in Figure 2 Applying the horizontal line test to each graph confirms the results we obtained in Example 1 FIGURE 2 Apply
11. 182 D 2 Graphs and Functions SECTION 2 6 e One to One Functions Inverse Functions e One to One Functions e Inverse Functions Many important mathematical relationships can be expressed in terms of functions For example C td f d The circumference of a circle is a function of the diameter d V g s The volume of a cube is a function of the edge s d 1 000 100p h p The demand for a product is a function of the price p 9 F zC 32 Temperature measured in F is a function of temperature in C In many cases we are interested in reversing the correspondence determined by a function Thus C d mC The diameter of a circle is a function of the circumference C T s W nV The edge of a cube is a function of the volume V 1 i p 10 Too r d The price of a product is a function of the demand d 5 et C gE 32 Temperature measured in C is a function of temperature in F As these examples illustrate reversing the relationship between two quantities often produces a new function This new function is called the inverse of the original func tion Later in this text we will see that many important functions for example log arithmic functions are actually defined as the inverses of other functions In this section we develop techniques for determining whether the inverse func tion exists some general properties of inverse functions and methods for finding the rule of corresponde
12. Vx 1 and y 4x x to determine if the result is the graph of a function and if the inverse of the original function exists FIGURE 4 5 y X2 3 y 2x 1 y x y gt Draw Inverse y y Draw Inverse y a b 188 2 Graphs and Functions FIGURE 5 Composition of f and fo Finding the inverse of a function defined by a finite set of ordered pairs is easy just reverse each ordered pair But how do we find the inverse of a function defined by an equation Consider the one to one function f defined by fx 2 1 To find f we let y f x and solve for x y 2x 1 y 1 2x sy 4 x Since the ordered pair x y is in f if and only if the reversed ordered pair y x is in f this last equation defines f x f Q py 3 1 Something interesting happens if we form the composition of f and f in either of the two possible orders x Nie F O f x 1 9x 1 2 5 and FE ON fay 2Gy l y 1 1 y These compositions indicate that if f maps x into y then f maps y back into x and if f maps y into x then f maps x back into y This is interpreted schematically in Figure 5 DOMAIN f RANGE f f D fa aD MQ y 1 RANGEf 1 i DOMAIN f t Finally we note that we usually use x to represent the independent variable and y the dependent variable in an equation that defines a function It is customary to do this for inverse functions also Thus interchanging the v
13. ariables x and y in equation 1 we can state that the inverse of y f 2x 1 is y f p43 When working with inverse functions it is customary to write compositions as f g x rather than as f e g x 2 6 Inverse Functions 189 In general we have the following result Theorem 5 Relationship between f and f If f exists then 1 x f y if and only if y f x 2 FI x for all x in the domain of f 3 FIFO y for all y in the domain of f or if x and y have been inter changed f f x x for all x in the domain of f If f and g are one to one functions satisfying flg x for all x in the domain of g gelf x for all x in the domain of f then it can be shown that g f and f g Thus the inverse function is the only function that satisfies both these compositions We can use this fact to check that we have found the inverse correctly EXPLORE DISCUSS 2 Find f g x and g f x for fa 1 2 and g x 2 8 1 How are f and g related The procedure for finding the inverse of a function defined by an equation is given in the next box This procedure can be applied whenever it is possible to solve y f x for x in terms of y Finding the Inverse of a Function f Step 1 Find the domain of f and verify that f is one to one If f is not one to one then stop since f does not exist Step 2 Solve the equation y f x for x The result is an equation o
14. b 2 a b 2 bisects the line segment from a b to b a a b see the figure In Problems 81 84 the function f is not one to one Find the inverses of the functions formed by restricting the domain of f as indicated Check Problems 81 84 by graphing f g and the line y x in a squared viewing window on a graphing utility Hint To restrict the graph of y f x to an interval of the form a Sx S b enter y f x K a S x x b 81 fx 2 x A x 2 B x 2 82 f x 1 x A xs 1 B x 1 83 f x V4x x A 0 lt x lt 2 84 f x Vox x A 0 lt x lt 3 B 2 lt x lt 4 B 3 lt x lt 6 198 2 Graphs and Functions CHAPTER 2 GROUP ACTIVITY Mathematical Modeling in Business This group activity is concerned with analyzing a basic model for manufacturing and selling a product by using tables of data and linear regression to determine appropriate values for the constants a b m and n in the fol lowing functions TABLE 1 Business Modeling Functions Function Definition Interpretation Price demand p x m nx x is the number of items that can be sold at p per item Cost C x a bx Total cost of producing x items Revenue R x xp Total revenue from the sale of x items x m nx Profit P x R x C x Total profit from the sale of x items A manufacturing company manufactures and sells mountain bikes The management would like to have price dema
15. dinate system Answers to Matched Problems 1 A Not one to one B One to one 2 f x 2 x20 3 fl 2 V4 x x4 lt 4 EXERCISE 2 6 A Which of the functions in Problems 1 16 are one to one 1 1 2 2 1 3 4 4 3 2 1 0 0 1 1 1 2 1 3 5 4 4 3 3 3 2 4 4 5 4 4 3 3 2 2 D 5 Domain Range 6 Domain Range 2 gt 4 2 ee 1 gt 2 ja o gt 0 o gt 7 1 gt _ 1 Teg 7 2 gt 5 ee 7 Domain Range 8 Domain Range L5 2 3 3 gt 1 4 _ gt 2 5 __ 4 10 11 12 13 2 6 Inverse Functions 195 gt X gt X i S m x A x A x A x Y Y gt X gt X 196 2 Graphs and Functions 14 3_ 9 4x xX ie 29 fi lt 30 fa A x 9 x 4 In Problems 31 34 use the graph of the one to one function f to sketch the graph of f State the domain and sh range of f 31 Spe 15 r x A oS gt X 32 y y x 5 gt X 16 s x A y f x Which of the functions in Problems 17 22 are one to one 17 F x 4x 2 19 Hix 4 x 21 M
16. f the form j O Step 3 Interchange x and y in the equation found in step 2 This expresses f as a function of x Step 4 Find the domain of f Remember the domain of f must be the same as the range of f 190 2 Graphs and Functions and Check your work by verifying that f lf x forall x in the domain of f fI x forall x in the domain of f EXAMPLE 2 Finding the Inverse of a Function Find f for f x Vx 1 Solution Step 1 gt lt Step 2 f x Vx 1 x21 FIGURE 6 Step 3 Step 4 Find the domain of f and verify that f is one to one The domain of f is 1 The graph of fin Figure 6 shows that fis one to one hence f exists Solve the equation y f x for x y Ve 1 y x 1 x y 1 Thus x f y 1 Interchange x and y yafl aetl Find the domain of f The equation f x x 1 is defined for all val ues of x but this does not tell us what the domain of f is Remember the domain of f must equal the range of f From the graph of f we see that the range of f is 0 Thus the domain of f is also 0 That is f x 1 x 0 Check For x in 1 the domain of f we have F IQ f Wx 1 Vx 17 1 x I14 1 IX X Matched Problem 2 2 6 Inverse Functions 191 For x in 0 the domain of f t we have FI O FO 1 VEFDAT Syp x Vx x for any real number x
17. function Now we want to see how we can form a new function by reversing the correspon dence determined by a given function Let g be the function defined as follows g 3 9 0 0 3 9 g is not oneto one Notice that g is not one to one because the domain elements 3 and 3 both corre spond to the range element 9 We can reverse the correspondence determined by func tion g simply by reversing the components in each ordered pair in g producing the following set G 9 3 0 0 9 3 G is not a function But the result is not a function because the domain element 9 corresponds to two dif ferent range elements 3 and 3 On the other hand if we reverse the ordered pairs in the function f 2 2 4 3 9 f is oneto one we obtain F 2 1 4 2 9 3 F is a function This time f is a one to one function and the set F turns out to be a function also This new function F formed by reversing all the ordered pairs in f is called the inverse of f and is usually denoted by f t Thus f 2 1 4 2 9 3 The inverse of f Notice that f is also a one to one function and that the following relationships hold Domain of f 2 4 9 Range of f Range of f t 1 2 3 Domain of f Thus reversing all the ordered pairs in a one to one function forms a new one to one function and reverses the domain and range in the process We are now ready to pre sent a formal definition of the
18. h the function CORE a ee on a graphing utility enter m 4x 0 A The Boolean expression x 0 is assigned the value 1 if the inequality is true and 0 if it is false How does this result in restricting the graph of x 4 to just those values of x satisfying x 0 B Use this concept to reproduce Figures 8 b and 8 c on a graphing utility C Do your graphs appear to be symmetric with respect to the line y x What happens if you use a squared window for your graph 2 6 Inverse Functions 193 Recall from Theorem 2 that increasing and decreasing functions are always one to one This provides the basis for a convenient and popular method of restricting the domain of a function If the domain of a function f is restricted to an interval on the x axis over which f is increasing or decreasing then the new function deter mined by this restriction is one to one and has an inverse We used this method to form the functions g and h in Figure 8 EXAMPLE 3 Finding the Inverse of a Function Find the inverse of f x 4x x x S 2 Graph f f and y x in the same coordinate system Solution Step 1 Find the domain of f and verify that f is one to one The graph of y 4x x is the parabola shown in Figure 9 a Restricting the domain of f to x 2 restricts the graph of f to the left side of this parabola Fig 9 b Thus f is a one to one function
19. ing the horizontal line test gt lt gt X gt X f x x does not pass f x 2x 1 passes the horizontal line test the horizontal line test is not one to one f is one to one b a A function that is increasing throughout its domain or decreasing throughout its domain will always pass the horizontal line test see Figs 3 a and 3 b Thus we have the following theorem Theorem 3 Increasing and Decreasing Functions If a function f is increasing throughout its domain or decreasing throughout its domain then f is a one to one function FIGURE 3 Increasing decreasing y and one to one functions A gt X A one to one function is not always increasing or decreasing c A decreasing function An increasing function is always one to one is always one to one a b 186 2 Graphs and Functions e Inverse Functions The converse of Theorem 3 is false To see this consider the function graphed in Figure 3 c This function is increasing on 0 and decreasing on 0 yet the graph passes the horizontal line test Thus this is a one to one function that is neither an increasing function nor a decreasing
20. nce that defines the inverse function A review of Section 2 3 will prove very helpful at this point Recall the set form of the definition of a function A function is a set of ordered pairs with the property that no two ordered pairs have the same first component and different second components However it is possible that two ordered pairs in a function could have the same sec ond component and different first components If this does not happen then we call the function a one to one function It turns out that one to one functions are the only functions that have inverse functions 2 6 Inverse Functions 183 DEFINITION 1 One to One Function A function is one to one if no two ordered pairs in the function have the same second component and different first components To illustrate this concept consider the following three sets of ordered pairs f 10 3 0 5 4 7 g 0 3 3 4 7 h 0 3 2 5 4 7 Set f is not a function because the ordered pairs 0 3 and 0 5 have the same first component and different second components Set g is a function but it is not a one to one function because the ordered pairs 0 3 and 2 3 have the same second com ponent and different first components But set h is a function and it is one to one Representing these three sets of ordered pairs as rules of correspondence provides some additional insight into this concept f g h
21. nd and cost functions for break even and profit loss analysis Price demand and cost functions may be established by collecting appropriate data at different levels of output and then finding a model in the form of a basic elementary function from our library of elementary functions that closely fits the collected data The financial department using statistical techniques arrived at the price demand and cost data in Tables 2 and 3 where p is the wholesale price of a bike for a demand of x thousand bikes and C is the cost in thou sands of dollars of producing and selling x thousand bikes TABLE 2 Price Demand TABLE 3 Cost x thousand p x thousand C thousand 7 530 5 2 100 13 360 12 2 940 19 270 19 3 500 25 130 26 3 920 A Building a Mathematical Model for Price Demand Plot the data in Table 2 and observe that the rela tionship between p and x is almost linear After observing a relationship between variables analysts often try to model the relationship in terms of a basic function from a portfolio of elementary functions which best fits the data 1 Linear regression lines are frequently used to model linear phenomena This is a process of fitting to a set of data a straight line that minimizes the sum of the squares of the distances of all the points in the graph of the data to the line by using the method of least squares Many graphing utilities have this routine built in Read your user
22. r coordinate system Interpret the slope and the y intercept of the cost function Discuss its domain and range Using the math ematical model determine the cost for an output and sales of 10 000 bikes For an output and sales of 20 000 bikes C Break Even and Profit Loss Analysis Write an equation for the revenue function and state its domain Write the equation for the profit function and state its domain 1 Graph the revenue function and the cost function simultaneously in the same rectangular coordinate system Algebraically determine at what outputs to the nearest unit the company breaks even Determine where costs exceed revenues and revenues exceed costs Graph the revenue function and the cost function simultaneously in the same viewing window Graphically determine at what outputs to the nearest unit the company breaks even and where costs exceed revenues and revenues exceed costs Graph the profit function in a rectangular coordinate system Algebraically determine at what outputs to the nearest unit the company breaks even Determine where profits occur and where losses occur At what out put and price will a maximum profit occur Do the maximum revenue and maximum profit occur for the same output Discuss Graph the profit function in a graphing utility Graphically determine at what outputs to the nearest unit the company breaks even and where losses occur and profits occur At what output and price will a maxi
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