1. Chapter 2.7 Function Operations and Composition

2. Arithmetic Operations on Functions As mentioned near the end of Section 2.3, economists frequently use the equation “profit equals revenue minus cost,” or P(x) = R(x) – C(x), where x is the number of items produced and sold.

3. That is, the profit function is fund by subtracting the cot function from the revenue function. Figure 94 shows the situation for a company that manufactures DVDs.

4. The two lines are the graphs of the linear functions for revenue R(x) = 168x and Cost C(x) = 118x +800 with x, R(x) and C(x) given in thousands

5. y x Example 1 Stretching or Shrinking a Graph Graph each function x|x| -2 0 1 2

6. When 30,000 DVDs are produced and sold, profit is P(30) = R(30) – C(30) = 168(30) – [118(30) + 800] = 5040 – 4340 = 700 That is the profit from the sale of 30,000 DVDs is $700

8. Example 1 Using Operations on Functions Let f(x) = x 2 + 1 and g(x) = 3x + 5 (a) (f + g)(1)

9. Example 1 Using Operations on Functions Let f(x) = x 2 + 1 and g(x) = 3x + 5 (b) (f - g)(-3)

10. Example 1 Using Operations on Functions Let f(x) = x 2 + 1 and g(x) = 3x + 5 (c) (f g)(5)

11. Example 1 Using Operations on Functions Let f(x) = x 2 + 1 and g(x) = 3x + 5

13. Example 2 Using Operations on Functions and Determining Domains

17. (e) Give the domains of the functions in parts (a) – (d).

18. Example 3 Evaluating Combinations of Functions If possible, use the given representations of functions f and g to evaluate

19. Example 3 Evaluating Combinations of Functions If possible, use the given representations of functions f and g to evaluate

20. Example 3 Evaluating Combinations of Functions If possible, use the given representations of functions f and g to evaluate

21. The Difference Quotient Suppose the point P lies on the graph of y = f(x), and h is a positive number. If we let (x, f(x)) denot the coordinates of P and (x+h, f(x+h)) denote the coordinates of Q, then the line joining P and Q has slope

22. This difference is called the difference quotient. Figure 96 shows the graph of the line PQ (called a secant line.

23. As h approaches 0, the slope of this secant line approaches the slope of the line tangent to the curve at P. Important applications of this idea are developed in calculus. The next example illustrates a three-step process for finding the difference quotient of a function.

24. Example 4 finding the Difference Quotient Let f(x) = 2x 2 – 3x. Find the difference quotient and simplify the expression. Step 1. Find f(x + h) Step 2. Find f(x + h) – f(x) Step 3. Find the difference quotient.

25. Composition of Functions The diagram in Figure 97 shows a function f that assigns to each x in its domain a value f(x).

26. Then another function g assigns to each f(x) in its domain a value g[f(x)]. This two step process takes an element x and produces a corresponding element g[f(x)].

29. As a real-life example of function composition, suppose an oil well off the California coast is leaking, with a leak spreading iol in a circular layer over the water’s surface.

30. At any time t, in minutes, after the beginning of the leak, the radius of the circular oil slick is r(t) = 5t feet.

33. Example 5 Evaluating Composite Functions

36. Example 6 Finding Composition Functions

39. Caution

40. Example 7 Finding Composite Functions and Their Domains

42. Example 8 Finding Functions That Form a Given Composite