Lesson 4-2 Operations on Functions Objective: To perform operations on functions and to determine the domain of the resulting functions.

Suppose a company manufactures and sells a certain product. If the cost of manufacturing x items of the product is given by the function C(x) and the revenue generated by the sale of the x items is given by the function R(x) then the company’s profit is given by the function P(x) where P(x) = R(x) - C(x) That is the profit function is the difference between the revenue and cost function. As the example from economics suggests it is possible to combine two given functions to produce a new function.

The Sum, Difference,Product and Quotient of Functions Each function listed below is defined for all x in the domain of both f and g. 1.Sum of f and g: (f +g)(x) = f(x) + g(x) 2.Difference of f and g: (f – g) (x) = f (x) - g (x) 3.Product of f and g: (f g) (x) = f(x) g(x) 4. Quotient of f and g

Example 1 Let f(x) = x + 1 and g(x) = x 2 -1 Find a rule for each of the following functions. a.(f + g)(x) = f(x) + g(x) = (x + 1) +(x 2 -1 ) = x 2 + x b.

A fifth way of combining functions can be illustrated by the sport of cycling. When an 18 speed touring bicycle is in sixth gear. The gear ratio is 3:2 which means that the wheel of the bicycle revolve 3 times for every 2 revolutions of the pedal. this relationship can be expressed as where w and p represent wheel and pedal revolutions, respectively. Since the wheels of a touring bicycle have a diameter of 27in. W revolutions of the wheels move the bicycle a distance d in inches, given by. d = 27πw

Notice that d = 27πw gives distance as a function of wheel revolutions and that w = 3 p 2 Gives wheel revolutions as a functions of pedal revolutions. By substituting 3 p for w in d = 27πw 2 We get d = 27π(3 p) = 40.5 πp 2 Which gives distance as a function of wheel revolutions. This function of d(p) is said to be a composite of the functions d(w) and w(p).

The composite of a Function The composite of f and g denoted f ⁰ g is denoted by two conditions: 1.(f ⁰ g)(x) = f(g(x)) which is read “f circle g of x equals f of g of x. 2.X is the domain of g and g and g(x) is the domain of f. The domain of f ⁰ g is the set of x satisfying condition (2) above. The operation that combine f and g produce their composite is called the composite of functions.

Example 2

Example 3

HOMEWORK 1-6,17-22 PGS Class Example: 1.) F(x) = 5 g(x) = 0 (f + g) (x) = = 5 b.) (f+g)(3) f(x) = 5 and g(x) = 3 (f+g) (x) = = 8

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