1. Functions 2 Copyright © Cengage Learning. All rights reserved.

2. 2.7 Combining Functions

3. 3 Objectives ■ Sums, Differences, Products, and Quotients ■ Composition of Functions ■ Applications of Composition

4. 4 Sums, Differences, Products, and Quotients

5. 5 Two functions f and g can be combined to form new functions f + g, f – g, fg, and f /g in a manner similar to the way we add, subtract, multiply, and divide real numbers. For example, we define the function f + g by (f + g) (x) = f (x) + g (x) The new function f + g is called the sum of the functions f and g; its value at x is f (x) + g (x). Of course, the sum on the right-hand side makes sense only if both f (x) and g (x) are defined, that is, if x belongs to the domain of f and also to the domain of g.

6. 6 Sums, Differences, Products, and Quotients So if the domain of f is A and the domain of g is B, then the domain of f + g is the intersection of these domains, that is, A  B. Similarly, we can define the difference f – g, the product fg, and the quotient f /g of the functions f and g. Their domains are A  B, but in the case of the quotient we must remember not to divide by 0.

7. 7 Example 1 – Combinations of Functions and Their Domains Let f (x) = and g (x) = (a) Find the functions f + g, f – g, fg, and f /g and their domains. (b) Find (f + g)(4), (f – g)(4), (fg)(4), and (f /g)(4). Solution: (a) The domain of f is {x | x  2}, and the domain of g is {x | x  0}. The intersection of the domains of f and g is {x | x  0 and x  2} = [0, 2)  (2, )

8. 8 Example 1 – Solution Thus, we have Note that in the domain of f /g we exclude 0 because g (0) = 0. Domain {x | x  0 and x  2} Domain {x | x > 0 and x  2} cont’d

9. 9 Example 1 – Solution (b) Each of these values exist because x = 4 is in the domain of each function. cont’d

10. 10 Sums, Differences, Products, and Quotients The graph of the function f + g can be obtained from the graphs of f and g by graphical addition. This means that we add corresponding y-coordinates, as illustrated in the next example.

11. 11 Example 2 – Using Graphical Addition The graphs of f and g are shown in Figure 1. Use graphical addition to graph the function f + g. Figure 1

12. 12 Example 2 – Solution We obtain the graph of f + g by “graphically adding” the value of f (x) to g (x) as shown in Figure 2. This is implemented by copying the line segment PQ on top of PR to obtain the point S on the graph of f + g. Graphical addition Figure 2

13. 13 Composition of Functions

14. 14 Composition of Functions Now let’s consider a very important way of combining two functions to get a new function. Suppose f (x) = and g (x) = x 2 + 1. We may define a new function h as h(x) = f (g (x)) = f (x 2 + 1) = The function h is made up of the functions f and g in an interesting way: Given a number x, we first apply the function g to it, then apply f to the result.

15. 15 Composition of Functions In this case, f is the rule “take the square root,” g is the rule “square, then add 1,” and h is the rule “square, then add 1, then take the square root.” In other words, we get the rule h by applying the rule g and then the rule f. Figure 3 shows a machine diagram for h. The h machine is composed of the g machine (first) and then the f machine. Figure 3

16. 16 Composition of Functions In general, given any two functions f and g, we start with a number x in the domain of g and find its image g (x). If this number g (x) is in the domain of f, we can then calculate the value of f (g (x)). The result is a new function h (x) = f (g (x)) that is obtained by substituting g into f. It is called the composition (or composite) of f and g and is denoted by f  g (“f composed with g”).

17. 17 Composition of Functions The domain of f  g is the set of all x in the domain of g such that g (x) is in the domain of f. In other words, (f  g) (x) is defined whenever both g (x) and f (g (x)) are defined. We can picture f  g using an arrow diagram (Figure 4). Arrow diagram for f  g Figure 4

18. 18 Example 3 – Finding the Composition of Functions Let f (x) = x 2 and g (x) = x – 3. (a) Find the functions f  g and g  f and their domains. (b) Find (f  g)(5) and (g  f )(7). Solution: (a) We have (f  g)(x) = f (g (x)) = f (x – 3) Definition of f  g Definition of g

19. 19 Example 3 – Solution = (x – 3) 2 and (g  f )(x) = g (f (x)) = g(x 2 ) = x 2 – 3 The domains of both f  g and g  f are Definition of f Definition of g  f Definition of f Definition of g cont’d

20. 20 Example 3 – Solution (b) We have (f  g)(5) = f (g (5)) = f (2) = 2 2 = 4 (g  f )(7) = g (f (7)) = g (49) = 49 – 3 = 46 cont’d

21. 21 Composition of Functions You can see from Example 3 that, in general, f  g  g  f. Remember that the notation f  g means that the function g is applied first and then f is applied second. It is possible to take the composition of three or more functions. For instance, the composite function f  g  h is found by first applying h, then g, and then f as follows: (f  g  h) (x) = f (g (h (x)))

22. 22 Example 5 – A Composition of Three Functions Find f  g  h if f (x) = x / (x + 1), g (x) = x 10, h (x) = x + 3. Solution: (f  g  h) (x) = f (g (h (x))) = f (g (x + 3)) = f ( (x + 3) 10 ) Definition of f  g  h Definition of h Definition of g Definition of f

23. 23 Composition of Functions So far, we have used composition to build complicated functions from simpler ones. But in calculus it is useful to be able to “decompose” a complicated function into simpler ones, as shown in the following example.

24. 24 Example 6 – Recognizing a Composition of Functions Given F (x) = find functions f and g such that F = f  g. Solution: Since the formula for F says to first add 9 and then take the fourth root, we let g (x) = x + 9 and f (x) = Then (f  g) (x) = f (g (x)) Definition of f  g

25. 25 Example 6 – Solution = f (x + 9) = F (x) Definition of g Definition of f cont’d

26. 26 Applications of Composition

27. 27 Applications of Composition When working with functions that model real-world situations, we name the variables using letters that suggest the quantity being modeled. We may use t for time, d for distance, V for volume, and so on. For example, if air is being pumped into a balloon, then the radius R of the balloon is a function of the volume V of air pumped into the balloon, say, R = f (V). Also the volume V is a function of the time t that the pump has been working, say, V = g (t). It follows that the radius R is a function of the time t given by R = f (g(t)).

28. 28 Example 7 – An Application of Composition of Functions A ship is traveling at 20 mi/h parallel to a straight shoreline. The ship is 5 mi from shore. It passes a lighthouse at noon. (a) Express the distance s between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon; that is, find f so that s = f (d). (b) Express d as a function of t, the time elapsed since noon; that is, find g so that d = g(t). (c) Find f  g. What does this function represent?

29. 29 Example 7 – Solution We first draw a diagram as in Figure 5. Figure 5

30. 30 Example 7 – Solution (a) We can relate the distances s and d by the Pythagorean Theorem. Thus s can be expressed as a function of d by (b) Since the ship is traveling at 20 mi/h, the distance d it has traveled is a function of t as follows: d = g(t) = 20t cont’d

31. 31 Example 7 – Solution (c) We have The function f  g gives the distance of the ship from the lighthouse as a function of time. Definition of g Definition of f Definition of f  g cont’d