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Chapter 1 Functions and Their Graphs Warm Up 1.5 Given f (x) = 2x – 3 and g(x) = x2 – 1, find: f (x) + g(x) = f (x) – g(x) = f (x) · g(x) = f (x) / g(x) = 2 1.5 Combinations of Functions Objectives: Add, subtract, multiply, and divide functions. Find compositions of one function with another function. Use combinations of functions to model and solve real-life problems. 3 Vocabulary Arithmetic Combination of Functions Composition of Functions 4 Arithmetic Operations Let f and g be two functions with overlapping domains. Then, for all x common to both domains: 1. 2. 3. (f + g)(x) = f (x) + g(x) (f – g)(x) = f (x) – g(x) (f g)(x) = f (x) · g(x) f f x 4. x , g x g g x 0 5 Example 1 Let f (x) = 2x +1 and g (x) = x2 + 2x – 1. Find (f – g)(x) algebraically and graphically. Then find (f – g)(2). 6 Example 2 Find (f /g)(x) and (g/f )(x) for the functions given by f x x and g x 2 4x . Specify the domain for each quotient. 7 Consider this … If you drop a pebble into a pond, a circular ripple extends out from the drop point. The radius of the circle is a function of time. The area enclosed by the circular ripple is a function of the radius. 8 What If …? Suppose that the radius is increasing at a constant rate of 8 in/sec. Then r = 8t What is the radius at t = 5? r = __________________ The formula for the area of the circular region is A = ____________________ What is the area at t = 5? A = ____________________ 9 How Are the Variables Related? Time is the input for the radius function. Time r(t) Radius A(r) or A(r(t)) Area And the radius is the input for the area function. 10 This means … Area is a function of time through this chain: Area depends on radius. Radius depends on time. Area is a composite function of time, A(r(t)). 11 Composition of Functions The composition of the function f with the function g is f g x f g x 12 Example 3 Find the following values of the using the values of f and g found in the table. a. Find f (g(2). b. Find g(f (2). x 1 2 3 4 5 6 f (x) 3 4 6 2 0 1 g (x) 5 3 2 1 7 4 Example 4 Let f ( x ) x , x 0 and g ( x ) x 1, x 1 . Find f g x . Then find, if possible, f g 2 and f g 0 . 14 Homework 1.5.1 Worksheet 1.5 #1, 7, 9, 17, 23, 35, 37, 51, 53, 73, 74 15 Warm Up 1.5 A square concrete foundation was prepared as a base for a large cylindrical gasoline tank as r shown. a. Write the radius r of the tank as a function of the length x of the sides of the square. b. Write the area A of the circular base of the tank as a function of the radius r. 16 In-Class Assignment Work with a partner to complete the worksheet “Domain of Composite Functions”. 17 Domain of Composite Functions The domain of f (g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The outputs of g must be restricted so that they are in the domain of f. 18 Finding the Domain Let f ( x ) 1 and g ( x ) x 1 . x Find the domain of f g x . The outputs of g can be any real number. But the domain of f is restricted to all real numbers ≠ 0. Therefore, g x 0 or x 1 . So, domain of f g is all real numbers except x 1 . 19 Example 4 Find the domain of the composition f g x for the functions given by f x x 9 and g x 2 9x . 2 20 Example 5 Does f g x g f x ? a. Let f x x 2 and g x 4 x 2 . b. Let f x 2 x 3 and g x 1 2 x 3 . 21 Special Case If (f ○g)(x) = (g ○ f)(x), then (f ○g)(x) = x and (g ○ f)(x) = x. 22 Application Problem 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 ˚C. When the food is removed from the refrigerator, 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)) & interpret its meaning. b. Find the number of bacteria in the food when t = 2 hours. c. Find the time when the bacterial count reaches 200. 23 Homework 1.5.2 Worksheet 1.5 # 39 – 49 odd, 63 – 71 odd, 77 24