ࡱ> A G  4bjbj ËËǨl8,J;YYYYbbbZ\\\\\\$`b bbbYY~bYYZbZYh /F0bbbbbbbbbbbbbbbbbbbbbbb : LESSON 4 COMPOSITION FUNCTIONS AND INVERSE FUNCTIONS 1. COMPOSTION FUNCTIONS Definiton Let f and g be two functions. The composite function  EMBED Equation.3  is the function defined by  EMBED Equation.3 . The domain of  EMBED Equation.3  is the set of all x in the domain of g such that  EMBED Equation.3  is in the domain of f.   .  EMBED Equation.3  g  f  x . Domain of f .  EMBED Equation.3   EMBED Equation.3  Domain of g E Example In HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3, we consider the horizontal shift of the function h given by  EMBED Equation.3 . This function is a composite function. Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  NOTE: The domain of g is the set of all real numbers. Since the domain of f is  EMBED Equation.3 , then  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . Thus, the domain of  EMBED Equation.3  is  EMBED Equation.3 . Example In HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson2/Lesson2.doc"Lesson 2, we found the domain of the function h given by  EMBED Equation.3 . This function is a composite function. Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  NOTE: The domain of g is the set of all real numbers x such that  EMBED Equation.3 . Since the domain of f is  EMBED Equation.3 , then  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . We found the solution to this non-linear inequality in Lesson 2 obtaining that  EMBED Equation.3 . Thus, the domain of  EMBED Equation.3  is  EMBED Equation.3 . Examples Given the function h, find functions f and g such that  EMBED Equation.3 . 1.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson2/Lesson2.doc"Lesson 2) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  2.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson2/Lesson2.doc"Lesson 2) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  3.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson2/Lesson2.doc"Lesson 2) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  4.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson2/Lesson2.doc"Lesson 2) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  5.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson2/Lesson2.doc"Lesson 2) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  6.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  7.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  8.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  9.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  10.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  OR Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3  11.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  OR Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3  12.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  13.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  OR Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3  14.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  15.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1850/Lectures/Lesson10/Lesson10.doc"Lesson 10 of HYPERLINK "http://www.math.utoledo.edu/~anderson/1850"MATH-1850) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  16.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1850/Lectures/Lesson10/Lesson10.doc"Lesson 10 of HYPERLINK "http://www.math.utoledo.edu/~anderson/1850"MATH-1850) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  17.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1850/Lectures/Lesson10/Lesson10.doc"Lesson 10 of HYPERLINK "http://www.math.utoledo.edu/~anderson/1850"MATH-1850) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  18.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1850/Lectures/Lesson10/Lesson10.doc"Lesson 10 of HYPERLINK "http://www.math.utoledo.edu/~anderson/1850"MATH-1850) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  19.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1850/Lectures/Lesson10/Lesson10.doc"Lesson 10 of HYPERLINK "http://www.math.utoledo.edu/~anderson/1850"MATH-1850) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  20.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1850/Lectures/Lesson10/Lesson10.doc"Lesson 10 of HYPERLINK "http://www.math.utoledo.edu/~anderson/1850"MATH-1850) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  21.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1850/Lectures/Lesson10/Lesson10.doc"Lesson 10 of HYPERLINK "http://www.math.utoledo.edu/~anderson/1850"MATH-1850) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  22.  EMBED Equation.3  (From HYPERLINK "http://www.math.utoledo.edu/~anderson/1850/Lectures/Lesson10/Lesson10.doc"Lesson 10 of HYPERLINK "http://www.math.utoledo.edu/~anderson/1850"MATH-1850) Let  EMBED Equation.3  and  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Answer:  EMBED Equation.3  and  EMBED Equation.3  2. INVERSE FUNCTIONS Example Consider the following function  EMBED Equation.3  and its inverse  EMBED Equation.3 .   EMBED Equation.3    x . . y   EMBED Equation.3  D E NOTE: The function  EMBED Equation.3  maps x in the set D to y in the set E and  EMBED Equation.3  maps y back to x. Of course, this is what an inverse function is suppose to do. Example Consider the following function  EMBED Equation.3 . Does  EMBED Equation.3  have an inverse? x . . a  y . . b . c D E The function  EMBED Equation.3  maps x in the set D to a in the set E. So, the inverse function would map a back to x. The function  EMBED Equation.3  maps y in the set D to b in the set E. So, the inverse function would map b back to y. However, where does the inverse function map c. Note that the domain of the function  EMBED Equation.3  is the set  EMBED Equation.3  and the range of the function  EMBED Equation.3  is the set  EMBED Equation.3 . The function  EMBED Equation.3  is not an onto function. In order for a function to be an onto function, every element in the set E must be used. For the given function  EMBED Equation.3 , the element c in the set E was not used. In order for a function to have an inverse, it must be an onto function. If a function is not an onto function, then the lack of this needed condition is easy to fix. To fix the lack of the onto condition, replace the set E by the range of the function.  EMBED Equation.3   x . . a  y . . b  EMBED Equation.3  D Range of  EMBED Equation.3  =  EMBED Equation.3  Now, this function has an inverse function. Notice the following relationship above: The domain of the inverse function  EMBED Equation.3  is equal to the range of the function  EMBED Equation.3 . Example Consider the following function  EMBED Equation.3 . Does  EMBED Equation.3  have an inverse?   x .  y . . c D E Note that the domain of the function  EMBED Equation.3  is the set D, which is the set  EMBED Equation.3  and the range of the function  EMBED Equation.3  is the set  EMBED Equation.3 . Also, note that the function  EMBED Equation.3  is an onto function. The set E is the range of the function  EMBED Equation.3 . The function  EMBED Equation.3  maps x in the set D to c in the set E. The function  EMBED Equation.3  also maps y in the set D to c in the set E. So, the inverse function would map c back to either x or y. Which one do you use? The problem here is that the function  EMBED Equation.3  is not a one-to-one function. In order for a function to be a one-to-one function, you may only use each element in the set E once. In order for a function to have an inverse, it must be a one-to-one function. If a function is not a one-to-one function, then the lack of this needed condition is not as easy to fix as the lack of the onto condition. In order to fix the lack of the one-to-one condition, you must put a restriction on the domain of the function. In other words, you must eliminate elements from the set D. What elements in the set D are you going to chose to eliminate? This is the reason that fixing the lack of the one-to-one condition is harder. For the function  EMBED Equation.3 , the domain is the set D =  EMBED Equation.3 . Thus, we will either eliminate x or y. Each restricted domain will produce an inverse function. Thus, these two choices for the restricted domain will produce two inverse functions. If we eliminate y, then we get the following inverse function:   EMBED Equation.3    x . . c   EMBED Equation.3  Restricted domain of  EMBED Equation.3  =  EMBED Equation.3  Range of  EMBED Equation.3  =  EMBED Equation.3  If we eliminate x, then we get the following inverse function:   EMBED Equation.3    y . . c   EMBED Equation.3  Restricted domain of  EMBED Equation.3  =  EMBED Equation.3  Range of  EMBED Equation.3  =  EMBED Equation.3  The function  EMBED Equation.3  has two possible inverse functions depending on the restricted domain that is chosen. Notice the following relationship for both inverse functions above: The restricted domain of the function  EMBED Equation.3  is equal to the range of the function  EMBED Equation.3 . Theorem A function  EMBED Equation.3  has an inverse function, denoted by  EMBED Equation.3  if and only if the function  EMBED Equation.3  is one-to-one and onto. We have the following relationships:   EMBED Equation.3    x . . y   EMBED Equation.3  Restricted domain of  EMBED Equation.3  Range of  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Range of  EMBED Equation.3  Domain of  EMBED Equation.3  We also have the following two relationships between a function and its inverse function: 1.  EMBED Equation.3  for all x in the restricted domain of  EMBED Equation.3  2.  EMBED Equation.3  for all y in the domain of  EMBED Equation.3  Example Find the inverse function of the function  EMBED Equation.3 . First, consider the graph of  EMBED Equation.3 .  y x Information about the domain of the function  EMBED Equation.3 can be determined by the x-coordinate of the points on the graph. Since the value of the x-coordinates range in value from negative infinity to positive infinity, then the domain of  EMBED Equation.3  is all real numbers. Information about the range of the function  EMBED Equation.3 can be determined by the y-coordinate of the points on the graph. Since the value of the y-coordinates range in value from zero to positive infinity, then the range of  EMBED Equation.3  is all real numbers greater than or equal to zero. In interval notation, we have the following: Domain of  EMBED Equation.3  =  EMBED Equation.3  and Range of  EMBED Equation.3  =  EMBED Equation.3  Since the range of  EMBED Equation.3  is  EMBED Equation.3 , then by the discussion above, the domain of  EMBED Equation.3  is  EMBED Equation.3 . Recall the following test for checking the graph of a function for being one-to-one: The Horizontal Line Test: If a horizontal line intersects the graph of a function in more than one place, then the function is not one-to-one. By the horizontal line test, the function  EMBED Equation.3  is not one-to-one. We will have to put a restriction on the domain of the function in order to fix this. We have the following two choices for the restricted domain: 1) the interval of numbers  EMBED Equation.3 ; that is, the set of numbers greater than or equal to zero or 2) the interval of numbers  EMBED Equation.3 ; that is, the set of numbers less than or equal to zero. Each restricted domain will produce an inverse function. Thus, these two choices for the restricted domain will produce two inverse functions. For the first choice of  EMBED Equation.3 , our restricted domain is  EMBED Equation.3  and the graph of the function  EMBED Equation.3  on the restricted domain looks like the following: y x Since the restricted domain of  EMBED Equation.3  is  EMBED Equation.3 , then by the discussion above, the range of  EMBED Equation.3  is  EMBED Equation.3 . For the second choice of  EMBED Equation.3 , our restricted domain is  EMBED Equation.3  and the graph of the function  EMBED Equation.3  on the restricted domain looks like the following:  y x Since the restricted domain of  EMBED Equation.3  is  EMBED Equation.3 , then by the discussion above, the range of  EMBED Equation.3  is  EMBED Equation.3 . Now, lets find the inverse function(s) algebraically. Set  EMBED Equation.3 :  EMBED Equation.3  Solve for x:  EMBED Equation.3  Thus, either  EMBED Equation.3  if the restricted domain of  EMBED Equation.3  is  EMBED Equation.3  or  EMBED Equation.3  if the restricted domain of  EMBED Equation.3  is  EMBED Equation.3 . Notice, as predicted above the domain of  EMBED Equation.3  is  EMBED Equation.3  and the range of  EMBED Equation.3  is  EMBED Equation.3  for the first inverse function and the range of  EMBED Equation.3  is  EMBED Equation.3  for the second inverse function. Now, lets verify the two relationships for this function and its two inverse functions are true. 1. For the first restricted of domain of  EMBED Equation.3  for the function  EMBED Equation.3 , we have that  EMBED Equation.3 : a.  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  = x. Note that since x belongs to the restricted domain of  EMBED Equation.3 , then x is a positive number. The absolute value of a positive number is itself. b.  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  = y . Note that since y is in the domain of  EMBED Equation.3  and the domain of  EMBED Equation.3  is  EMBED Equation.3 , then y is greater than or equal to zero. Thus, the square root of x is defined as a real number. 2. For the second restricted of domain of  EMBED Equation.3  for the function  EMBED Equation.3 , we have that  EMBED Equation.3 : a.  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  = x . Note that since x belongs to the restricted domain of  EMBED Equation.3 , then x is a negative number. The absolute value of a negative number is the negative of itself. b.  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  = y . Note that since y is in the domain of  EMBED Equation.3  and the domain of  EMBED Equation.3  is  EMBED Equation.3 , then y is greater than or equal to zero. Thus, the square root of x is defined as a real number. Thus, the two relationships for this function and its two inverse functions hold. Examples Find the inverse function for the following functions. State the domain and range of both the function and its inverse. 1.  EMBED Equation.3 ,  EMBED Equation.3  The graph of the function  EMBED Equation.3  was sketched in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3.  y x You can see from this sketch that the graph fails the Horizontal Line Test. Thus, the function  EMBED Equation.3  is not one-to-one. Here is the sketch of the graph of the function for  EMBED Equation.3 .  y x This graph does pass the Horizontal Line Test. Thus, the function  EMBED Equation.3  for  EMBED Equation.3  is one-to-one. Restricted Domain of f =  EMBED Equation.3  = Range of  EMBED Equation.3  Range of f =  EMBED Equation.3  = Domain of  EMBED Equation.3  Now, we will find  EMBED Equation.3 . Setting  EMBED Equation.3 , we have that  EMBED Equation.3 . Now solving for x, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . Since  EMBED Equation.3 , then  EMBED Equation.3 . That is,  EMBED Equation.3  is nonnegative. Thus,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . Thus,  EMBED Equation.3 . Since the domain of  EMBED Equation.3  is the range of f, then find the range of f by finding the domain of  EMBED Equation.3 . For the domain of  EMBED Equation.3 , we want that  EMBED Equation.3 . Thus,  EMBED Equation.3   EMBED Equation.3 . The domain of  EMBED Equation.3  is  EMBED Equation.3 . Thus, the range of f is  EMBED Equation.3 . This is the same answer that we obtained in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3 using the sketch of the graph of the function f. Answer:  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  The graphs of the functions  EMBED Equation.3  for  EMBED Equation.3  and  EMBED Equation.3  are given below. The graph of  EMBED Equation.3  is red and the graph of  EMBED Equation.3  is blue:  y x Notice that the graph of  EMBED Equation.3  is a reflection of the graph of  EMBED Equation.3  through the line  EMBED Equation.3 , which is gray. 2.  EMBED Equation.3 ,  EMBED Equation.3  The graph of the function  EMBED Equation.3  was sketched in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3.  y x You can see from this sketch that the graph fails the Horizontal Line Test. Thus, the function  EMBED Equation.3  is not one-to-one. Here is the sketch of the graph of the function for  EMBED Equation.3 .  y x This graph does pass the Horizontal Line Test. Thus, the function  EMBED Equation.3  for  EMBED Equation.3  is one-to-one. Restricted Domain of f =  EMBED Equation.3  = Range of  EMBED Equation.3  Range of f =  EMBED Equation.3  = Domain of  EMBED Equation.3  Now, we will find  EMBED Equation.3 . Setting  EMBED Equation.3 , we have that  EMBED Equation.3 . Now solving for x, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . Since  EMBED Equation.3 , then  EMBED Equation.3 . That is,  EMBED Equation.3  is not positive. Thus,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . Thus,  EMBED Equation.3 . Since the domain of  EMBED Equation.3  is the range of f, then find the range of f by finding the domain of  EMBED Equation.3 . For the domain of  EMBED Equation.3 , we want that  EMBED Equation.3 . Thus,  EMBED Equation.3   EMBED Equation.3 . The domain of  EMBED Equation.3  is  EMBED Equation.3 . Thus, the range of f is  EMBED Equation.3 . This is the same answer that we obtained in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3 using the sketch of the graph of the function f. Answer:  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  The graphs of the functions  EMBED Equation.3  for  EMBED Equation.3  and  EMBED Equation.3  are given below. The graph of  EMBED Equation.3  is red and the graph of  EMBED Equation.3  is blue:  y x Notice that the graph of  EMBED Equation.3  is a reflection of the graph of  EMBED Equation.3  through the line  EMBED Equation.3 , which is gray. 3.  EMBED Equation.3 ,  EMBED Equation.3  The graph of the function  EMBED Equation.3  was sketched in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3.  y x You can see from this sketch that the graph fails the Horizontal Line Test. Thus, the function  EMBED Equation.3  is not one-to-one. Here is the sketch of the graph of the function for  EMBED Equation.3 .  y x This graph does pass the Horizontal Line Test. Thus, the function  EMBED Equation.3  for  EMBED Equation.3  is one-to-one. Restricted Domain of g =  EMBED Equation.3  = Range of  EMBED Equation.3  Range of g =  EMBED Equation.3  = Domain of  EMBED Equation.3  Now, we will find  EMBED Equation.3 . Setting  EMBED Equation.3 , we have that  EMBED Equation.3 . Now solving for x, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . Since  EMBED Equation.3 , then  EMBED Equation.3 . That is,  EMBED Equation.3  is nonnegative. Thus,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . Thus,  EMBED Equation.3 . Since the domain of  EMBED Equation.3  is the range of g, then find the range of g by finding the domain of  EMBED Equation.3 . For the domain of  EMBED Equation.3 , we want that  EMBED Equation.3 . Thus,  EMBED Equation.3   EMBED Equation.3 . The domain of  EMBED Equation.3  is  EMBED Equation.3 . Thus, the range of g is  EMBED Equation.3 . This is the same answer that we obtained in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3 using the sketch of the graph of the function g. Answer:  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  The graphs of the functions  EMBED Equation.3  for  EMBED Equation.3  and  EMBED Equation.3  are given below. The graph of  EMBED Equation.3  is red and the graph of  EMBED Equation.3  is blue: y x Notice that the graph of  EMBED Equation.3  is a reflection of the graph of  EMBED Equation.3  through the line  EMBED Equation.3 , which is gray. 4.  EMBED Equation.3 ,  EMBED Equation.3  The graph of the function  EMBED Equation.3  was sketched in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3.  y x You can see from this sketch that the graph fails the Horizontal Line Test. Thus, the function  EMBED Equation.3  is not one-to-one. Here is the sketch of the graph of the function for  EMBED Equation.3 .  y x This graph does pass the Horizontal Line Test. Thus, the function  EMBED Equation.3  for  EMBED Equation.3  is one-to-one. Restricted Domain of g =  EMBED Equation.3  = Range of  EMBED Equation.3  Range of g =  EMBED Equation.3  = Domain of  EMBED Equation.3  Now, we will find  EMBED Equation.3 . Setting  EMBED Equation.3 , we have that  EMBED Equation.3 . Now solving for x, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . Since  EMBED Equation.3 , then  EMBED Equation.3 . That is,  EMBED Equation.3  is not positive. Thus,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Thus,  EMBED Equation.3 . Since the domain of  EMBED Equation.3  is the range of g, then find the range of g by finding the domain of  EMBED Equation.3 . For the domain of  EMBED Equation.3 , we want that  EMBED Equation.3 . Thus,  EMBED Equation.3   EMBED Equation.3 . The domain of  EMBED Equation.3  is  EMBED Equation.3 . Thus, the range of g is  EMBED Equation.3 . This is the same answer that we obtained in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3 using the sketch of the graph of the function g. Answer:  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  The graphs of the functions  EMBED Equation.3  for  EMBED Equation.3  and  EMBED Equation.3  are given below. The graph of  EMBED Equation.3  is red and the graph of  EMBED Equation.3  is blue: y x Notice that the graph of  EMBED Equation.3  is a reflection of the graph of  EMBED Equation.3  through the line  EMBED Equation.3 , which is gray. 5.  EMBED Equation.3  The graph of this function was sketched in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3.  y x This graph passes the Horizontal Line Test. Thus, the function h is one-to-one. Domain of h =  EMBED Equation.3  = Range of  EMBED Equation.3  Range of h =  EMBED Equation.3  = Domain of  EMBED Equation.3  Now, we will find  EMBED Equation.3 . Setting  EMBED Equation.3 , we have that  EMBED Equation.3 . Now solving for x, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Thus,  EMBED Equation.3 . Since the domain of  EMBED Equation.3  is the range of h, then find the range of h by finding the domain of  EMBED Equation.3 . The domain of  EMBED Equation.3  is the set of all real numbers. Thus, the range of h is the set of all real numbers. This is the same answer that we obtained in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3 using the sketch of the graph of the function h. Answer:  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  The graphs of the functions  EMBED Equation.3  and  EMBED Equation.3  are given below. The graph of  EMBED Equation.3  is red and the graph of  EMBED Equation.3  is blue:  y x Notice that the graph of  EMBED Equation.3  is a reflection of the graph of  EMBED Equation.3  through the line  EMBED Equation.3 , which is gray. 6.  EMBED Equation.3  The graph of this function was sketched in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3.  y t This graph passed the Horizontal Line Test. Thus, the function f is one-to-one. Domain of f =  EMBED Equation.3  = Range of  EMBED Equation.3  Range of f =  EMBED Equation.3  = Domain of  EMBED Equation.3  Now, we will find  EMBED Equation.3 . Setting  EMBED Equation.3 , we have that  EMBED Equation.3 . Now solving for t, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Thus,  EMBED Equation.3 . NOTE:  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Thus,  EMBED Equation.3 . Since the domain of  EMBED Equation.3  is the range of f, then find the range of f by finding the domain of  EMBED Equation.3 . The domain of  EMBED Equation.3  is the set of all real numbers. Thus, the range of f is the set of all real numbers. This is the same answer that we obtained in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3 using the sketch of the graph of the function f. Answer:  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3  The graphs of the functions  EMBED Equation.3  and  EMBED Equation.3  are given below. The graph of  EMBED Equation.3  is red and the graph of  EMBED Equation.3  is blue:  y t Notice that the graph of  EMBED Equation.3  is a reflection of the graph of  EMBED Equation.3  through the line  EMBED Equation.3 , which is gray. 7.  EMBED Equation.3  The graph of this function was sketched in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3. y x This graph passes the Horizontal Line Test. Thus, the function g is one-to-one. Domain of g =  EMBED Equation.3  = Range of  EMBED Equation.3  Range of g =  EMBED Equation.3  = Domain of  EMBED Equation.3  Now, we will find  EMBED Equation.3 . Setting  EMBED Equation.3 , we have that  EMBED Equation.3 . Now solving for x, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . Since the domain of  EMBED Equation.3  is  EMBED Equation.3 , then  EMBED Equation.3 , where  EMBED Equation.3 . Answer:  EMBED Equation.3 ,  EMBED Equation.3  or  EMBED Equation.3 ,  EMBED Equation.3  The graphs of the functions  EMBED Equation.3  and  EMBED Equation.3  for  EMBED Equation.3  are given below. The graph of  EMBED Equation.3  is red and the graph of  EMBED Equation.3  is blue:  y x Notice that the graph of  EMBED Equation.3  is a reflection of the graph of  EMBED Equation.3  through the line  EMBED Equation.3 , which is gray. 8.  EMBED Equation.3  The graph of this function was sketched in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3. y t This graph passes the Horizontal Line Test. Thus, the function h is one-to-one. Domain of h =  EMBED Equation.3  = Range of  EMBED Equation.3  Range of h =  EMBED Equation.3  = Domain of  EMBED Equation.3  Now, we will find  EMBED Equation.3 . Setting  EMBED Equation.3 , we have that  EMBED Equation.3 . Now solving for x, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Since the domain of  EMBED Equation.3  is  EMBED Equation.3 , then  EMBED Equation.3 , where  EMBED Equation.3 . Answer:  EMBED Equation.3 ,  EMBED Equation.3  or  EMBED Equation.3 ,  EMBED Equation.3  The graphs of the functions  EMBED Equation.3  and  EMBED Equation.3  for  EMBED Equation.3  are given below. The graph of  EMBED Equation.3  is red and the graph of  EMBED Equation.3  is blue:  y t Notice that the graph of  EMBED Equation.3  is a reflection of the graph of  EMBED Equation.3  through the line  EMBED Equation.3 , which is gray. 9.  EMBED Equation.3  The graph of this function was sketched in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3.  y  EMBED Equation.3  x This graph passes the Horizontal Line Test. Thus, the function h is one-to-one. Domain of function =  EMBED Equation.3  = Range of inverse function Range of function =  EMBED Equation.3  = Domain of inverse function Now, we will find the inverse function. Solving for x in  EMBED Equation.3 , we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Since the domain of the inverse function is  EMBED Equation.3 , then  EMBED Equation.3 , where  EMBED Equation.3 . Answer:  EMBED Equation.3 ,  EMBED Equation.3  or  EMBED Equation.3 ,  EMBED Equation.3  The graphs of the function  EMBED Equation.3  and its inverse function  EMBED Equation.3  for  EMBED Equation.3  are given below. The graph of  EMBED Equation.3  is red and the graph of  EMBED Equation.3  for  EMBED Equation.3  is blue: y x Notice that the graph of the inverse function is a reflection of the graph of the function  EMBED Equation.3  through the line  EMBED Equation.3 , which is gray. 10.  EMBED Equation.3  The graph of this function was sketched in HYPERLINK "http://www.math.utoledo.edu/~anderson/1320/Lectures/Lesson3/Lesson3.doc"Lesson 3.  y x This graph passes the Horizontal Line Test. Thus, the function h is one-to-one. Domain of f =  EMBED Equation.3  = Range of  EMBED Equation.3  Range of f =  EMBED Equation.3  = Domain of  EMBED Equation.3  Now, we will find  EMBED Equation.3 . Setting  EMBED Equation.3 , we have that  EMBED Equation.3 . Now solving for x, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Thus,  EMBED Equation.3 . Answer:  EMBED Equation.3  or  EMBED Equation.3  The graphs of the functions  EMBED Equation.3  and  EMBED Equation.3  are given below. The graph of  EMBED Equation.3  is red and the graph of  EMBED Equation.3  is blue:  y x Notice that the graph of  EMBED Equation.3  is a reflection of the graph of  EMBED Equation.3  through the line  EMBED Equation.3 , which is gray. 11.  EMBED Equation.3 ,  EMBED Equation.3  12.  EMBED Equation.3 ,  EMBED Equation.3  13.  EMBED Equation.3      Copyrighted by James D. 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