Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 13. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | CHAP E R Functions Working Together In this chapter we ll look at ways of combining functions in order to construct new functions. COMBINING OUTPUTS ADDITION SUBTRACTION MULTIPLICATION AND DIVISION OF FUNCTIONS The sum difference product and quotient of functions are the new functions defined respectively by the addition subtraction multiplication and division of the outputs or values of the original functions. Addition and Subtraction of Functions EXAMPLE Suppose a company produces The revenue money the company takes in by selling widgets is a function of x where x is the number of widgets produced. We call this function R x . We call C x the cost of producing x widgets. Producing and selling x widgets results inaprofit P x where profit is revenue minus cost P x R x - C x . The height of the graph of the profit function is obtained by subtracting the height of the cost function from the height of the revenue function. Where P x is negative the company loses money. The x-intercept of the P x graph corresponds to the break-even point where revenue exactly equals costs so there is zero profit. See Figure on the following page. 1A widget is an imaginary generic product frequently used by economists when discussing hypothetical companies. 101 102 CHAPTER 3 Functions Working Together Figure More generally if h is the sum of functions and g h x x g x then the output of h corresponding to an input of x1 is the sum of x1 and g x1 . In terms of the graphs the height of h at x1 is the sum of the heights of the graphs of and g at x1. An analogous statement can be made for subtraction. The domain of h is the set of all x common to the domains of both and g. EXAMPLE Let x x and g x 1. We are familiar with the graphs of and g. We can obtain a rough sketch of x g x x 1 from the graphs of and g by adding together the values of the functions as shown in the figure below. Figure EXERCISE The following questions refer to Example where x g x x
