Section 1.5 17 Combinations of Functions Course Number Section 1.5 Combinations of Functions Instructor Objective: In this lesson you learned how to find arithmetic combinations and compositions of functions. I. Arithmetic Combinations of Functions (Pages 51−53) Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other Date What you should learn How to add, subtract, multiply, and divide functions real numbers, two functions f and g can be combined to create new functions such as the quotient sum, difference, product, and of f and g to create new functions. 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. Let f and g be two functions with overlapping domains. Complete the following arithmetic combinations of f and g for all x common to both domains: 1) Sum: ( f + g )( x) = f(x) + g(x) 2) Difference: 3) Product: 4) Quotient: ( f − g )( x) = ( fg )( x) = §f · ¨¨ ¸¸(x) = ©g¹ f(x) − g(x) f(x) · g(x) f(x) ÷ g(x), g(x) ≠ 0 To use a graphing utility to graph the sum of two functions, . . . enter the first function as y1 and enter the second function as y2. Define y3 as y1 + y2, then graph y3. Example 1: Let f ( x) = 7 x − 5 and g ( x) = 3 − 2 x . Find ( f − g )(4) . 28 Larson/Hostetler/Edwards Precalculus: Functions and Graphs, A Graphing Approach, 5th Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. 18 Chapter 1 II. Compositions of Functions (Pages 53−56) The composition of the function f with the function g is ( f D g )( x) = f(g(x)) . Functions and Their Graphs What you should learn How to find compositions of one function with another function For the composition of the function f with g, the domain of f D g is . . . the set of all x in the domain of g such that g(x) is in the domain of f. Example 2: Let f ( x) = 3x + 4 and let g ( x) = 2 x 2 − 1 . Find (a) ( f D g )( x) and (b) ( g D f )( x) . (a) 6x2 + 1 (b) 18x2 + 48x + 31 III. Applications of Combinations of Functions (Page 57) The function f ( x) = 0.06 x represents the sales tax owed on a purchase with a price tag of x dollars and the function g ( x) = 0.75 x represents the sale price of an item with a price tag of x dollars during a 25% off sale. Using one of the combinations of functions discussed in this section, write the function that represents the sales tax owed on an item with a price tag of x dollars during a 25% off sale. What you should learn How to use combinations of functions to model and solve real-life problems (f g)(x) = (g f)(x) = 0.045x Additional notes Homework Assignment Page(s) Exercises Larson/Hostetler/Edwards Precalculus: Functions and Graphs, A Graphing Approach, 5th Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved.