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1 _CH04_p3-369.pdf Page Algebra With Functions The basic rules of algebra tell you how the operations of addition and multiplication behave. Addition and multiplication are operations that combine numbers. In this lesson, you will learn about an operation that combines functions. Here are two functions. B(p) is the birthday of person p. The domain is all people, and the target is the days of the year. W(d ) is the day of the week on which a date d will occur next year. The domain is the days of the year, and the target is the seven days of the week. You can compose these two functions to form a third function. C(p) is the day of the week of person p s birthday next year. The domain is all people, and the target is the seven days of the week. C is a composition of functions W and B. To find C(p), you start with the person, find that person s birthday, and then find the day of the week of that date next year. In other words, C is the result of running an input through B, taking the output, and running that output through W. Find each value. 1. B(you). W(you) 3. C(you) You can express the relationship between these functions with the equation C(p) = W(B(p)) This tells you how to compute the output of C for any input. If you just want to say that C is the composition of W and B, you write it as C = W + B. Composing functions together by using the output of one as the input of the next is very common. You can do this with any two functions for which the domain of one function contains the target of another. If f i A S B and g i B S C, you can define a new function that maps inputs in A to outputs in C with the following rule. x A g( f (x)) This new function has domain A and target C. Instead of using a new letter to name it, you can just call the composition g + f. You read g + f as g circle f or g composed with f. So B(Carl Gauss) 5 April 30. When you evaluate g(f(x)), first you do f and then you do g Algebra With Functions 83

2 _CH04_p3-369.pdf Page 53 Developing Look for relationships. You can think of functions as objects on which you can perform operations, just as you do with numbers. g + f is a function. g( f (x)) is the output of that function for input x. Composition takes two functions and produces a third function. The concept of an operation with two inputs and one output is very common in algebra. Composition of functions is another example. The addition and the multiplication of two numbers are two more examples of a binary operation. Some of the concepts that apply to addition and multiplication may carry over to composition. Here are some questions to consider. Addition and multiplication have identity elements. You can add 0 to a number and it does not change. You can multiply a number by 1 and it does not change. Is there an identity function for composition? Addition and multiplication have inverses (with one exception). All numbers have opposites that sum to 0, the identity. All numbers other than 0 have reciprocals. Do compositions of functions have inverses? Addition and multiplication are commutative. You can add or multiply in either order: = and 3? 5 = 5? 3. Is composition commutative? When you work on these questions, you are thinking about functions as objects rather than machines. Composition is an operation on these objects, just as addition and multiplication are operations on numbers. Here is the formal definition of composition. Definition For two functions f : A m B and g : B m C, the composite function g + f meets the following conditions. g + f : A m C g + f (x) 5 g(f (x)) Operations that take two inputs and produce one output are binary operations. They are functions, too, as long as they always produce the same output given the same two inputs. Artist s rendition of f + g 84 Chapter 4 Functions

3 _CH04_p3-369.pdf Page 54 Suppose f, g, hi R S R and f (x) = x 1 g(x) = x 1 1 h(x) = 4x 3 3x 4. Build models for the three functions in your function-modeling language. 5. Show that f + g(5) = Find g + f (5), f + h(5), and h + f (5). 7. Find formulas for f + g and g + f. Are these functions the same? Historical Perspective Algebra began as a collection of methods for finding unknown numbers for solving equations. The object was to find general formulas, like the quadratic formula, that provided recipes for solving a class of equations. Mathematicians stated the recipes in terms of the operations of arithmetic. (Take the negative of the coefficient of x, add it to the square root of the square of that coefficient minus...) The goal was to make the recipe independent of the actual numbers in the equations. Over time, mathematicians began to look for similar formulas in systems other than numbers that had different operations. Gradually, the focus shifted from the formulas to the operations themselves. This systems approach to algebra led people to investigate properties of operations and to make lists of the properties that are useful in calculating properties that are very similar to the basic rules of arithmetic. Algebra today deals with all kinds of systems numbers, polynomials, matrices, functions, and more exotic objects. Each system has one or more operations that allow for calculations. One such system is the set of functions, all with the same domain and target, in which the operation is composition. The notation f, g, h ;R S R means that f, g, and h all have domain R and target R. The output is dependent on the input, but the recipe itself is independent of both Algebra With Functions 85

4 _CH04_p3-369.pdf Page 55 Check Your Understanding 1. Consider the functions f (x) = x 1 3 and g(x) = 5x 1 1. Find each value. a. f (3) b. g(3) c. f (g(3)) d. f + g(3) e. g + f (3) f. f (3)? g(3). Let f (x) = x 1 3 and g(x) = 5x 1 1. a. Find a formula for g + f (x). b. Find a formula for f + g(x). 3. Suppose you drop a stone into a pond. It makes concentric circular ripples. The radius of a ripple as a function of time is r = 4t, where r is the radius in inches and t is the time in seconds. a. Express the area of a ripple as a function of its radius. b. Express the area of a ripple as a function of time. 4. Suppose f i R S R and f (x) = x. For each definition of g, find formulas for f + g(x) and g + f (x). a. g(x) = x 1 3 b. g(x) = x 7 c. g(x) = (x 4) 3 d. Write About It Explain what it might mean to say that the function f is the identity function on R. 5. Suppose f (x) = 3x 1. If possible, find a function g such that f + g(x) = x. 6. Suppose f (x) = x 1 3. If possible, find a function g such that g + f (x) = x. 7. Suppose f (x) = x 1 5. If possible, find a function g that makes each equation true. a. f + g(x) = 4x 1 1 b. g + f (x) = 4x Take It Further Let f (x) = ax 1 b and g(x) = cx 1 d. a. Find formulas for f + g(x) and g + f (x). b. Find conditions on a, b, c, and d that make f + g = g + f. 9. Take It Further Find a linear function f (x) = ax 1 b such that f + f (x) = 4x 1 9. Unless otherwise stated, all functions in these exercises have domain R and target R. Go nline 86 Chapter 4 Functions

5 _CH04_p3-369.pdf Page 56 On Your Own 10. Consider the functions f (x) = x 1 and g(x) = 3x 1 1. Find each value. a. f (4) b. g(4) c. f (4)? g(4) d. f (g(4)) e. f + g(4) f. g + f (4) 11. Suppose hi R S R and h(x) = "x. Show that h = a, where a(x) = u x u. 1. Suppose f (x) = x 5x 1 6 and g(x) = x. Find each value. a. f + g(3) b. g + f (3) c. f + g(a) d. g + f (a) e. ( f + g) + f (a) f. f + (g + f )(a) g. Find all numbers a such that f + g(a) = 0. h. Find all numbers a such that g + f (a) = Use the functions below. f (x) = x 6x 1 8 g(x) = x 1 3 h(x) = x 1 1 Find a formula for each composition. a. h + ( f + g)(x) b. (h + f ) + g(x) 14. Suppose f (x) = x 10x 1 1. If possible, find linear functions g and h that make each equation true. a. f + g(x) = x 4 b. h + ( f + g)(x) = x 15. If f (x) = x, find a function g, that is not equal to f, such that f + g = g + f. 16. Standardized Test Prep Suppose f(x) = x 5. Find a function g such that g + f(x) = 3x 11x 0. A. g(x) = 3x 1 19x B. g(x) = 3x 1 4 C. g(x) = (3x 4) D. g(x) = (3x 1 4) Go nline A linear function is a function in the form x A ax 1 b for some numbers a and b. Why do you call it linear? 4.08 Algebra With Functions 87

6 _CH04_p3-369.pdf Page Consider these three functions. Find a formula for each composition. a. a + b b. b + c c. (a + b) + c d. a + (b + c) a(x) = 3x 1 1 b(x) = x 7 c(x) = x Suppose a, b, c i R S R. Show that (a + b) + c = a + (b + c). 19. Take It Further Suppose f (x) = x 1 3 and g(x) = x. Find a way to construct the graph of f + g from the graphs of f and g. Maintain Your Skills 0. For each function f, find a function g such that f + g(x) = x. a. f (x) = x 1 3 b. f (x) = x 3 c. f (x) = 3x 1 5 d. f (x) = 3x 5 e. f (x) = x 1 5 f. f(x) = Ax 1 B, where A 0 1. For each function f, find a function g such that g + f (x) = x. a. f (x) = x 1 3 b. f (x) = x 3 c. f (x) = 3x 1 5 d. f (x) = 3x 5 e. f (x) = x 1 5 f. f(x) = Ax 1 B, where A 0 In arithmetic, addition is associative. For any numbers a, b, and c, (a 1 b) 1 c = a 1 (b 1 c). Go nline Video Tutor 88 Chapter 4 Functions

7 _CH04_p3-369.pdf Page Inverses: Doing and Undoing Suppose you have an output and you want to find the input that generated it. Can you always find it? In-Class Experiment Have everyone think of an integer from 1 to 0. Then, follow these steps: Subtract 10 from your number. Square the result. Add 7 to the squared result to get an ending number. List everyone s starting and ending numbers. 1. Did any students get the same ending number? More important, did any two students who got the same ending number start with different numbers?. If a student gives you an ending number, can you always find the starting number? Explain. Try the experiment again. This time, cube the number in the second step. 3. Did any students get the same ending number this time? Did any two students who got the same ending number start with different numbers? 4. If a student gives you an ending number, can you always find the starting number? Explain. Minds in Action Sasha and Derman are looking at the results from the In-Class Experiment. Derman Tony and Michelle got the same number at the end, and they started with different numbers: Tony had 13 and Michelle had 7. Sasha Maybe we should look at what they did with their numbers. Tony Michelle Starting number After subtracting 10 After squaring After adding Inverses: Doing and Undoing 89

8 _CH04_p3-369.pdf Page 59 Derman Either way, they end up with 16. I wonder if we can retrace the steps from the output 16. Ending number After undoing addition by 7 After undoing squaring Tony 16 9??? Michelle Derman Hmm. We can t undo the squaring. If x = 9, then x could be 3 or3. Sasha It looks like there s no way to decide. If you only told me Tony and Michelle ended up with 16, I couldn t figure out what their original numbers were. Derman It s weird. That didn t happen the second time we ran the experiment. If two people started with different numbers, they always got different results. That time, Tony started with 5 and Michelle started with 1. Sasha Starting number After subtracting 10 After cubing After adding 7 Tony Michelle What happens if we try to undo the operations? Ending number After undoing addition by 7 After undoing cubing After undoing subtraction by 10 Tony ??? Michelle Derman No problems this time. We found their starting numbers. I think you could do this with any ending number. Sasha To go from the ending number back to the starting number, we should start with the last operation. So we have these steps: Subtract seven from the ending number. Take the cube root of that. Add ten to that. We will be able to get back the starting number! Derman As long as all the steps can be undone. I wonder how we can decide whether or not a step can be undone Why can Sasha and Derman undo cubing but not squaring? 90 Chapter 4 Functions

9 _CH04_p3-369.pdf Page In the second experiment, why can Sasha and Derman always derive the starting number from the ending number? 6. Change the first experiment in some way to make its process reversible. The second In-Class Experiment amounted to evaluating the function f (x) = (x 10) for different starting values of x. When your class evaluated this function in the In-Class Experiment, you found that, if two students got the same output, they must have started with the same input. When a function has this behavior, it is one-to-one. Definition A function is one-to-one if its ouputs are unique. That is, a function f is one-to-one if f(r) = f(s) only when r = s. The function from the first In-Class Experiment is not one-to-one, since the different inputs 13 and 7 both give the same output, 16. That is, f (13) = f (7). But Which of these functions are one-to-one? P(x) = x Q(x) = x 3 R(x) = u x 10u S(x) =!x 1 15 You can sometimes tell that a function from R to R is not one-to-one from its graph. Here is the graph of a function g : R S R. From the graph, you can see that there are different inputs that give the same output. If two points on the graph have the same y-height, they correspond to different inputs that produce the same output. This graph shows two different inputs a and b with g(a) = g(b). If a horizontal line crosses the graph of a function in more than one place, the function cannot be one-to-one. y O a y b Establish a process. To check if a function f is one-to-one, assume that f(r) = f(s) and try to show that r = s. Is P one-to-one? That is, if P(r) = P(s), does r have to equal s? If r = s, does r = s? No, not necessarily! Be careful. The graph of an R-to-R function cannot tell you with certainty that a function is one-to-one, because you can never see the complete graph. For example, suppose f(x) = x 3 x and g(x) = f(x 0). Graph g over the interval 10 # x # 10. No horizontal line cuts the graph twice. But is g oneto-one? Check out the graph over the interval 10 # x # Inverses: Doing and Undoing 91

10 _CH04_p3-369.pdf Page 61 Use graphs to determine which functions are not one-to-one. Which functions are one-to-one? Explain. 8. A(x) = x 9. B(x) = x 10. C(x) = x D(x) = 1 x 1. E(x) = u xu 13. F(x) =!x 14. G(x) = x 3 x 15. H(x) = x 3 1 x Developing Visualize. Here is a potato-and-arrow diagram for a one-to-one function. One important fact about one-to-one functions is that they are reversible, as Derman and Sasha noticed. In the diagram, if you pick any output in Set B, it is always possible to determine where it came from. This means there is another function, an inverse function, from Set B to Set A. In a potato-and-arrow diagram, the inverse function looks similar, but all the arrows reverse direction. A You will see the formal definition of an inverse function in a moment. Derman and Sasha used inverse functions in their dialog. The function add 7 has an inverse: subtract 7. The function cube the number has an inverse: take the cube root. The function subtract 10 has an inverse: add 10. It might appear that the squaring and square root functions are inverses, but they are not. When the domain is R, the squaring function is not oneto-one, so the squaring step cannot be inverted. What do you find if the domain is different? B A B Any two arrows from A end at different objects in B. Only a one-toone function has this property. The domain and range switch. If you follow an arrow in the original function and then the corresponding arrow of the inverse function, you are back where you started. See Exercise 4. 9 Chapter 4 Functions

11 _CH04_p3-369.pdf Page 6 You can write the inverse of f as f 1. Here is the formal definition. Definition Suppose f is a one-to-one function with domain A and range B. The inverse function f 1 is a function with these properties: f 1 has domain B and range A. For all x in B, f (f 1 (x)) 5 x. There is another way to state this definition that puts the emphasis on the function rather than on its output. The identity function on a set is the function id that simply returns its input. Definition id(x) = x (Alternate Version) Suppose f is a one-to-one function with domain A and range B. The inverse function f 1 is a function with these properties: f 1 has domain B and range A. For all x in B, f + f 1 5 id. Derman s description of the function f (x) = (x 10) refers to f as a composition of three simpler functions, in this order. Subtract 10. Cube. Add 7. Sasha and Derman then describe how to recover the original input from the output. Subtract 7. Take the cube root. Add 10. This process describes the inverse function of f (x) = (x 10) You can build the inverse function by starting with x and applying the three reversing rules in the order above. f 1 (x) = " 3 x When an inverse function exists, you can often find it by describing the steps to reverse the process. There is an identity function for each domain, but they all behave the same way: They do nothing to an input. What is the graph of id : R S R? The original function f is the composition f = c + b + a of the steps listed. Then the inverse f 1 is f 1 = a 1 + b 1 + c 1. You invert each function and reverse their order Inverses: Doing and Undoing 93

12 _CH04_p3-369.pdf Page 63 Example Problem Find the inverse function of f (x) = Solution Developing x 1 5. Method 1 One way to do this is to describe f (x)as steps: Divide by and then add 5. The inverse function f 1 takes an input, subtracts 5, and then multiplies by. f 1 (x) = (x 5) Method The other way to find an inverse function is to use the definition. f (f 1 (x)) = x If f (x) = x 1 5, then f ( f 1 f (x)) = 1 (x) 1 5 Since x = f ( f 1 (x)), you have f 1 (x) x = 1 5 Solve for f 1 (x) as you would for any variable. You can subtract 5 from each side and then multiply by. x = f 1 (x) 1 5 x 5 = 16. Find the inverse of g(x) = (x 5). f 1 (x) (x 5) = f 1 (x) 17. What happens when you try to find the inverse of h(x) = (x 10) 1 7? Look for relationships. In arithmetic, numbers have additive inverses. When you add a number to its additive inverse, you get 0, the identity for addition. The additive inverse of a number is its opposite. Nonzero numbers have multiplicative inverses. When you multiply a number by its multiplicative inverse, you get 1, the identity for multiplication. The multiplicative inverse of a number is its reciprocal. Represent a function. You can think of this equation as f(anything) = anything Chapter 4 Functions

13 _CH04_p3-369.pdf Page 64 One-to-one functions have inverses with respect to composition. When you compose a function with its inverse, you get id, the identity for composition. There is one hitch: composition, unlike addition and multiplication, is not commutative. f + g g + f unless f and g are special functions. For a function and its inverse, though, the story is much simpler, thanks to the following theorem. Theorem 4.5 Proof Suppose f : A m B is one-to-one. Then f 1 : B m A is one-to-one A f 1 B 1 5 f Suppose f 1 (r) = f 1 (s). Take f of each side to conclude that r s. For f to be the inverse of f 1 (that is, f = ( f 1 ) 1 ), f must have a domain A and range B (which it does). Also, f 1 (f (a))must equal a for all a in A. You know f + f 1 is the identity, so f (f 1 (f (a)) = f + f 1 (f (a)) = id(f (a) = f (a). The fact that f is one-to-one and f ( f 1 (f (a)) = f (a) means that f 1 ( f (a)) = a. Thus f fits the definition of the inverse of f 1. So, f = ( f 1 ) 1 ). Check Your Understanding 1. The basic functions below are all from R to R. For each function, determine whether the function has an inverse. If it does, find the inverse function. If it does not, explain why not. a. f (x) = x b. g(x) = 1 x c. h(x) = x d. k(x) = x 3 e. /(x) = x 3 x f. m(x) =!x g. n(x) = u xu The functions f + g and g + f do not necessarily have the same domain. Detect the key characteristics. Study the two characteristics of f proved in the second part. They meet the requirements for f to be the inverse of f 1. That is why you can conclude f = (f 1 ) Inverses: Doing and Undoing 95

14 _CH04_p3-369.pdf Page 65. Use the graph of a function and its inverse at the right. a. The point (1, 5) is on the graph of the function. What point must be on the graph of the inverse function? b. If f i R S R, describe how you get the graph of f 1 from the graph of f. c. How does this graph connect to the statement of Theorem 4.5? 3. Use the definition of f below. f (x) = e x if x, 0 x if 0 # x # 6 a. What is the natural domain of f? b. Sketch the graph of f. c. Extend the definition of f so that its domain is all of R and f is one-to-one. d. Extend the definition of f so that its domain is all of R and f is not one-to-one. 4. The function x A x is not one-to-one on its natural domain. a. Restrict its domain to a set on which the function is one-to-one. b. On this restricted domain, what is the inverse of x A x? 5. The table at the right defines the function t. a. What is the domain of t? b. Draw a potato-and-arrow diagram that illustrates t(x). c. Why is t a function? d. Does t have an inverse? If so, give the table for t. If not, change the table to make a new function that is one-to-one. 6. Find the inverse of f (x) = x x 1, where x Functions h and j are defined on R 1. h(x) = x Graph h(x) and j(x). Is h equal to j 1? j(x) =!x 6 4 O x x y t(x) (1, 5) This is the function from Exercise 9 in Lesson In exercises like this, the table is the entire function. If an input is not in the table, then it is not in the domain of t. 96 Chapter 4 Functions

15 _CH04_p3-369.pdf Page Suppose f, g : R S R, with f (x) = 5x 17x 1 6 and g (x) = 5x. a. Find a formula for g + f (x). b. Find a formula for h(x) = g + f + g 1 (x). c. Draw the graphs of f and h on the same axes. d. Find the zeros of h and the zeros of f. 9. Take It Further Prove or disprove the following statement. If f (x) = f 1 (x) for an input x, then f (x) = x. On Your Own 10. Here are the graphs of four functions. Which functions are definitely not one-to-one? a. b. y 10 4 O 4 x c. y d. 1 O 4 6 8x 11. Standardized Test Prep Function f (x) is one-to-one, with f (3) = 7 and f (7) = 5. Which equation must be true? A. f 1 (3) = 1 7 B. f 1 (3) = 7 C. f 1 (7) = 5 D. f 1 (7) = 3 4 O 4x 1. Prove that, if functions f, g : R S R are one-to-one, then f + g is one-to-one x 4 O 4 8 y y The zeros of a function j are the numbers a that make j(a) = 0. Visualize. Some of these functions may be one-to-one. Think about how the graph may extend for x. 4 if the function really is one-to-one. How might it look if the function is not one-to-one? Go nline Inverses: Doing and Undoing 97

16 _CH04_p3-369.pdf Page Functions f and g are one-to-one. Which function is the inverse function of f + g? A. f 1 + g 1 B. g 1 + f 1 C. f 1 + g D. g 1 + f 14. Find the inverse of each function. a. m(x) = 5x 1 3 b. n(x) = x 11 c. p(x) = 3x 1 4 d. q(x) = x The graph of the function f (x) = ax 1 b is a line with slope a. a. If a 0, show that f is one-to-one. b. Find a formula for f 1 (x). c. Find the slope of the graph of the inverse function f 1. d. Find three linear functions g, h, and j such that each is its own inverse. 16. Suppose f, g, k : R S R, f (x) = 5x 3 1x 11x 1 6, g(x) = 5x, and k(x) = 5x. a. Find a formula for g + f (x). b. Find a formula for h(x) = g + f + k 1 (x). c. Draw the graphs of f and h on the same axes. d. Find the zeros of f and the zeros of h. 17. Take It Further Suppose f : R S R and f (x) = 7x 15x 1. Find a linear function g such that g + f + g 1 is a monic quadratic polynomial. 18. Write About It The graph of function f : R S R is increasing, which means that as x increases, f (x) also increases. a. Draw a possible graph of function f. b. Roy says that any increasing function, no matter what its graph looks like, must be one-to-one. Do you agree? Explain. Maintain Your Skills 19. Let f (x) = 4x 1 3. Find each value. a. f (10) b. f 1 (43) c. f (f (0)) d. f 1 ( f 1 (15)) e. f ( f 1 (89)) f. f 1 ( f (16.3)) g. f ( f 1 ( f (10))) The domain of each function is R. A quadratic is monic if the coefficient of x is Let f (x) = 4x 1 3. Find each value in terms of r. a. f (r) b. f ( f (r)) c. f ( f ( f (r))) d. f ( f ( f ( f (r)))) e. f ( f ( f ( f ( f (r))))) f. f 1 (r) The notation f 1 (r) means compose 1 copies of f and apply the composition to r. 98 Chapter 4 Functions