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Pre Calculus

Pre Calculus. Day 8. Plan for the day. Review Homework 1.5 page 56-57 # 29-35 odd, 43, 49, 53-62 all 1.6 page 65 # 9-12 all, 19-45 odd Combinations of Functions Adding Subtracting Multiplying Division Composite Inverse Functions Homework 5

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Pre Calculus

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  1. Pre Calculus Day 8

  2. Plan for the day • Review Homework • 1.5 page 56-57 # 29-35 odd, 43, 49, 53-62 all • 1.6 page 65 # 9-12 all, 19-45 odd • Combinations of Functions • Adding • Subtracting • Multiplying • Division • Composite • Inverse Functions • Homework 5 • 1.7 Pages 74-76 # 5, 9, 11, 13, 15, 19, 21, 35, 37, 47-55 odd • 1.8 Pages 83-85 # 1-4, 25, 29, 31, 39-51 odd (do not graph), 69-77 odd

  3. COMBINATIONS OF FUNCTIONS 1.7

  4. What You Should Learn • Add, subtract, multiply, and divide functions. • Find the composition of one function with another function. • Use combinations and compositions of functions to model and solve real-life problems.

  5. Arithmetic Combinations of Functions

  6. Arithmetic Combinations of Functions Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to create new functions. For example, the functions given by f(x)= 2x – 3 and g(x) = x2 – 1 can be combined to form the sum, difference, product, and quotient of f and g. f(x) + g(x) = (2x – 3) + (x2 – 1) = x2 + 2x – 4 Sum

  7. Arithmetic Combinations of Functions f(x) – g(x) = (2x – 3) – (x2 – 1) = –x2 + 2x – 2 f(x)g(x) = (2x – 3)(x2 – 1) = 2x3 – 3x2– 2x + 3 Difference Product Quotient

  8. Arithmetic Combinations of Functions – Domain Considerations • The domain of an arithmetic combination of functions f and g consists of all real numbers that are common to the domains of f and g. • In the case of the quotient f(x)/g(x), there is the further restriction that g(x)  0.

  9. Arithmetic Combinations of Functions

  10. Example – Finding the Sum of Two Functions Given f(x) = 2x + 1 and g(x) = x2 + 2x – 1 find (f + g)(x). Then evaluate the sum when x = 3. Solution: (f + g)(x) = f(x) + g(x) = (2x + 1) + (x2 + 2x – 1) When x = 3, the value of this sum is (f + g)(3) = 32 + 4(3) = x2 + 4x = 21.

  11. Composition of Functions

  12. Composition of Functions Another way of combining two functions is to form the composition of one with the other. For instance, if f(x) = x2 and g(x) = x + 1, the composition of f with g is f(g(x)) = f(x + 1) = (x + 1)2. This composition is denoted as fg and reads as “f composed with g.”

  13. Composition of Functions

  14. Example – Composition of Functions Given f(x) = x + 2 and g(x) = 4 – x2, find the following. a. (f g)(x) b. (g f)(x) c. (g f)(–2) Solution: a. The composition of f with g is as follows. (f g)(x) = f(g(x)) = f(4– x2) = (4– x2) + 2 = –x2 + 6 Definition of f g Definition of g(x) Definition of f(x) Simplify.

  15. Example – Solution cont’d b. The composition of g with f is as follows. (g f)(x) = g(f(x)) = g(x + 2) = 4 – (x + 2)2 = 4 – (x2 + 4x + 4) = –x2 – 4x Note that, in this case, (f g)(x)  (g f)(x). Definition of g f Definition of f(x) Definition of g(x) Expand. Simplify.

  16. Example – Solution cont’d c. Using the result of part (b), you can write the following. (g f )(–2) = –(–2)2 – 4(–2) = –4 + 8 = 4 Substitute. Simplify. Simplify.

  17. Composition of Functions In the previous example, you formed the composition of two given functions. In calculus, it is also important to be able to identify two functions that make up a given composite function. For instance, the function h given by h(x) = (3x – 5)3 is the composition of f with g where f(x) = x3 and g(x) = 3x – 5. That is, h(x) = (3x – 5)3 = [g(x)]3 = f(g(x)).

  18. Composition of Functions Basically, to “decompose” a composite function, look for an “inner” function and an “outer” function. In the function h, g(x) = 3x – 5 is the inner function and f(x) = x3 is the outer function.

  19. Application

  20. Example– Bacteria Count The number N of bacteria in a refrigerated food is given by N(T)= 20T2 – 80T + 500, 2 T  14 where T is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by T(t) = 4t + 2, 0 t  3 where t is the time in hours. a) Find the composition N(T(t)) and interpret its meaning in context. b) Find the time when the bacteria count reaches 2000.

  21. Example (a) – Solution N(T(t))= 20(4t + 2)2 – 80(4t + 2)+ 500 = 20(16t2 + 16t + 4) – 320t – 160 + 500 = 320t2 + 320t + 80 – 320t – 160 + 500 = 320t2 + 420 The composite function N(T(t)) represents the number of bacteria in the food as a function of the amount of time the food has been out of refrigeration.

  22. Example (b) – Solution cont’d The bacteria count will reach 2000 when 320t2 + 420 = 2000. Solve this equation to find that the count will reach 2000 when t 2.2 hours. When you solve this equation, note that the negative value is rejected because it is not in the domain of the composite function.

  23. Practice Given: Find:

  24. Solution

  25. Solution:

  26. Solution:

  27. Solution - One option …

  28. Another option …

  29. 1.8 INVERSE FUNCTIONS

  30. What You Should Learn • Find inverse functions informally and verify that two functions are inverse functions of each other. • Use graphs of functions to determine whether functions have inverse functions. • Use the Horizontal Line Test to determine if functions are one-to-one. • Find inverse functions algebraically.

  31. Inverse Functions

  32. Inverse Functions Recall that a function can be represented by a set of ordered pairs. For instance, the function f(x) = x + 4 from the set A = {1, 2, 3, 4} to the set B = {5, 6, 7, 8} can be written as follows. f(x) = x + 4: {(1, 5), (2, 6), (3, 7), (4, 8)} In this case, by interchanging the first and second coordinates of each of these ordered pairs, you can form the inverse function of f, which is denoted by f–1.

  33. Inverse Functions It is a function from the set B to the set A, and can be written as: f–1(x) = x – 4: {(5, 1), (6, 2), (7, 3), (8, 4)} Note that the domain of fis equal to the range of f–1,and vice versa, as shown below. Also note that the functions f and f–1 have the effect of “undoing” each other.

  34. Inverse Functions In other words, when you form the composition of f with f–1 or the composition of f–1 with f, you obtain the identity function. f(f–1(x)) = f(x – 4) = (x – 4) + 4 = x f–1(f(x)) = f–1(x + 4) = (x +4) – 4 = x

  35. Example – Finding Inverse Functions Informally Find the inverse of f(x) = 4x. Then verify that both f(f–1(x)) and f–1(f(x)) are equal to the identity function. Solution: The function fmultiplieseach input by 4. To “undo” this function, you need to divideeach input by 4. So, the inverse function of f(x) = 4x is

  36. Example 1 – Solution cont’d You can verify that both f(f–1(x)) = x and f–1(f(x)) = x as follows.

  37. Definition of Inverse Function

  38. Definition of Inverse Function Do not be confused by the use of –1 to denote the inverse function f–1. In this text, whenever f–1 is written, it alwaysrefers to the inverse function of the function f and notto the reciprocal of f(x). If the function g is the inverse function of the function f, it must also be true that the function f is the inverse function of the function g. For this reason, you can say that the functions f and g are inverse functions of each other.

  39. The Graph of an Inverse Function

  40. The Graph of an Inverse Function The graphs of a function f and its inverse function f–1 are related to each other in the following way. If the point (a, b) lies on the graph of f, then the point (b, a) must lie on the graph of f–1, and vice versa. This means that the graph of f–1 is a reflection of the graph of f in the line y = x,as shown.

  41. Example – Finding Inverse Functions Graphically Sketch the graphs of the inverse functions f(x) = 2x – 3 and on the same rectangular coordinate system and show that the graphs are reflections of each other in the line y = x.

  42. Example – Solution The graphs of f and f–1 are shown below. It appears that the graphs are reflections of each other in the line y = x. You can further verify this reflective property by testing a few points on each graph.

  43. Example – Solution cont’d Note in the following list that if the point (a, b) is on the graph of f, the point (b, a) is on the graph of f–1. Graph of f(x) = 2x – 3 Graph of (–1, –5) (–5, –1) (0, –3) (–3, 0) (1, –1) (–1, 1) (2, 1) (1, 2) (3, 3) (3, 3)

  44. One-to-One Functions

  45. One-to-One Functions The reflective property of the graphs of inverse functions gives you a nice geometric test for determining whether a function has an inverse function. This test is called the Horizontal Line Test for inverse functions.

  46. One-to-One… For a function to be one-to-one: For every x in the domain, there is only one y in the range and For every y in the range, there is only one x in the domain

  47. One-to-One Functions If no horizontal line intersects the graph of f at more than one point, then no y-valueis matched with more than one x-value. This is the essential characteristic of what are called one-to-one functions.

  48. One-to-One Functions Consider the function given by f(x) = x2. The table on the left is a table of values for f(x) = x2. The table of values on the right is made up by interchanging the columnsof the first table.

  49. One-to-One Functions The table on the right does not represent a function because the input x = 4 is matched with two different outputs: y = –2 and y = 2. So, f(x) = x2 is not one-to-one and does not have an inverse function.

  50. Example – Applying the Horizontal Line Test The graph of the function given by f(x) = x3 – 1 is shown below. Because no horizontal line intersects the graph of f at more than one point, you can conclude that f is a one-to-one function and doeshave an inverse function.

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