# Combinations of Functions; Composite Functions

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Combinations of Functions; Composite Functions

Objectives Students will be able to add, subtract, multiply and divide functions. Students will be able to find the composition of one function with another function. Students will be able to use combinations and composition of functions to model and solve real life problems.

Definitions: Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions. The sum of f + g, the difference f – g, the product fg, and the quotient f /g are functions whose domains are the set of all real numbers common to the domains of f and g, defined as follows: Sum: (f + g)(x) = f (x)+g(x) Difference: (f – g)(x) = f (x) – g(x) Product: (f • g)(x) = f (x) • g(x) Quotient: (f / g)(x) = f (x)/g(x), provided g(x) does not equal 0

Example Let f(x) = 2x+1 and g(x) = x2-2. Find f + g, f - g, fg, and f/g and state the domain of each.. Solution: f+g = 2x+1 + x2-2 = x2+2x-1 f-g = (2x+1) - (x2-2)= -x2+2x+3 fg = (2x+1)(x2-2) = 2x3+x2-4x-2 f/g = (2x+1)/(x2-2) (f+g)(2) = (2)2+2(2)-1 = 7

The Composition of Functions
The composition of the function f with g is denoted by f o g and is defined by the equation (f o g)(x) = f (g(x)). The domain of the composite function f o g is the set of all x such that x is in the domain of g and g(x) is in the domain of f. See pg. 105 figure P.96

Example Given f (x) = 2x – 8 and g(x) = x2 - 5, find: a.(f o g)(x)
b. (g o f)(x) What is the domain of each? Try Page 107 # 12, 16, 26, 28, 38, 42 Homework: Page 107 – 108 #9, 13, 15, 17, 21, 25, 27, 37 – 47 0dd