Chapter 4 Lesson 2 Combining Functions; Composite Functions

Operations with Functions OperationFormula 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

Revenue, Cost, And Profit  Equation for revenue: R(x) = px  Where p is the price per unit and x is the number of units produced  Equation for cost: C(x) = V + F  Where v is the variable cost and f is the fixed cost  Variable cost grows when more units are produced  Fixed cost remains the same regardless of how many units are produced  Equation for profit: P(x) = R(x) – C(x)

Revenue, Cost, and Profit  The demand for a certain electronic component is given by p(x) = x. Producing and selling x units of this component involve a monthly fixed cost of $1999, and the cost of producing each component is $4.  Write equations that model total revenue and total cost as functions of the units produced for this electronic component in a month.  Write the equation that models the profit as a function of the units produced for this component during the month.  Find the maximum monthly profit.

Peanut Production  Georgia’s production of peanuts has increased moderately in the last 10 years, but profit margins have been reduced by lower prices, declining yields, and increasing costs. The function that models the revenue from peanut production in Georgia for the years is given by R(x) = x thousand dollars, where x is the number of years after The function that models the total cost of peanut production in Georgia is given by C(x) = x thousand dollars, where x is the number of years after  Write the equation P(x) that models the profit for peanut production.  Graph R, C, and P on the same axes  What is the slope of P(x)? Interpret the slope as a rate of change  If the model is accurate, what will be the profit in 2010?

Average Cost  Form the average cost function if the total cost function for the production of x units of a product is C(x) = 50x  For which input values is this function defined? Give a real- world explanation of this answer.

Composition of Functions  The composite function, f of g, is denoted f ◦ g and defined by (f◦g)(x)=f(g(x))  The domain of f ◦ g is the subset of the domain of g for which f ◦ g is defined  The composite function g ◦ f is defined by (g ◦ f)(x) = g(f(x))  The domain of g ◦ f is the subset of the domain of f for which g ◦ f is defined

Examples

Combinations with Functions

Homework  Pages  1-25 odd,29,33,37,42,43,45,47,50,52