333202_0108.qxd 12/7/05 8:43 AM Page 84 84 Chapter 1 Functions and Their Graphs 1.8 Combinations of Functions: Composite Functions What you should learn •Add,subtract, multiply, and Arithmetic Combinations of Functions divide functions. Just as two real numbers can be combined by the operations of addition, subtrac- •Find the composition of tion, multiplication, and division to form other real numbers, two functions can be one function with another combined to create new functions. For example, the functions given by function. f ͑x͒ ϭ 2x Ϫ 3 and g͑x͒ ϭ x 2 Ϫ 1 can be combined to form the sum, difference, •Use combinations and compo- product, and quotient of f and g. sitions of functions to model and solve real-life problems. f ͑x͒ ϩ g͑x͒ ϭ ͑2x Ϫ 3͒ ϩ ͑x 2 Ϫ 1͒ Why you should learn it ϭ x 2 ϩ 2x Ϫ 4 Sum Compositions of functions can f ͑x͒ Ϫ g͑x͒ ϭ ͑2x Ϫ 3͒ Ϫ ͑x 2 Ϫ 1͒ be used to model and solve ϭϪ 2 ϩ Ϫ real-life problems. For instance, x 2x 2 Difference in Exercise 68 on page 92, f ͑x͒g͑x͒ ϭ ͑2x Ϫ 3͒͑x 2 Ϫ 1͒ compositions of functions are used to determine the price of ϭ 2x 3 Ϫ 3x 2 Ϫ 2x ϩ 3 Product a new hybrid car. f ͑x͒ 2x Ϫ 3 ϭ , x ±1 Quotient g͑x͒ x2 Ϫ 1 The domain of an arithmetic combination of functions f and g consists of all real numbers that are common to the domains of f and g. In the case of the quo- tient f͑x͒͞g͑x͒, there is the further restriction that g͑x͒ 0. Sum, Difference, Product, and Quotient of Functions Let f and g be two functions with overlapping domains. Then, for all x © Jim West/The Image Works common to both domains, the sum, difference, product, and quotient of f and g are defined as follows. 1. Sum: ͑ f ϩ g͒͑x͒ ϭ f ͑x͒ ϩ g͑x͒ 2. Difference: ͑ f Ϫ g͒͑x͒ ϭ f ͑x͒ Ϫ g͑x͒ 3. Product: ͑ fg͒͑x͒ ϭ f ͑x͒ и g͑x͒ f f ͑x͒ 4. Quotient:΂ ΃͑x͒ ϭ , g͑x͒ 0 g g͑x͒ Example 1 Finding the Sum of Two Functions Given f ͑x͒ ϭ 2x ϩ 1 and g͑x͒ ϭ x 2 ϩ 2x Ϫ 1, find ͑ f ϩ g͒͑x͒. Solution ͑ f ϩ g͒͑x͒ ϭ f ͑x͒ ϩ g͑x͒ ϭ ͑2x ϩ 1͒ ϩ ͑x 2 ϩ 2x Ϫ 1͒ ϭ x 2 ϩ 4x Now try Exercise 5(a). 333202_0108.qxd 12/7/05 8:43 AM Page 85 Section 1.8 Combinations of Functions: Composite Functions 85 Example 2 Finding the Difference of Two Functions Additional Examples Given f ͑x͒ ϭ 2x ϩ 1 and g͑x͒ ϭ x 2 ϩ 2x Ϫ 1, find ͑ f Ϫ g͒͑x͒. Then evaluate a. Given f ͑x͒ ϭ x ϩ 5 and g͑x͒ ϭ 3x, the difference when x ϭ 2. find ͑fg͒͑x͒. Solution Solution ͑fg͒͑x͒ ϭ f ͑x͒ и g͑x͒ The difference of f and g is ϭ ͑ ϩ ͒͑ ͒ x 5 3x ͑ f Ϫ g͒͑x͒ ϭ f ͑x͒ Ϫ g͑x͒ ϭ 3x2 ϩ 15x ϭ ͑2x ϩ 1͒ Ϫ ͑x 2 ϩ 2x Ϫ 1͒ ͑ ͒ ϭ 1 ͑ ͒ ϭ x b. Given f x and g x ϩ , x x 1 ϭϪx 2 ϩ 2. find ͑gf͒͑x͒. When x ϭ 2, the value of this difference is Solution 2 ͑gf͒͑x͒ ϭ g͑x͒ и f ͑x͒ ͑ f Ϫ g͒͑2͒ ϭϪ͑2͒ ϩ 2 x 1 ϭ ΂ ΃΂ ΃ ϭϪ2. x ϩ 1 x 1 Now try Exercise 5(b). ϭ , x 0 x ϩ 1 In Examples 1 and 2, both f and g have domains that consist of all real numbers. So, the domains of ͑ f ϩ g͒ and ͑ f Ϫ g͒ are also the set of all real numbers. Remember that any restrictions on the domains of f and g must be considered when forming the sum, difference, product, or quotient of f and g. Example 3 Finding the Domains of Quotients of Functions f g Find ΂ ΃͑x͒ and ΂ ΃͑x͒ for the functions given by g f f ͑x͒ ϭ Ίx and g͑x͒ ϭ Ί4 Ϫ x 2 . Then find the domains of f͞g and g͞f. Solution The quotient of f and g is f f ͑x͒ Ίx ΂ ΃͑x͒ ϭ ϭ g g͑x͒ Ί4 Ϫ x 2 and the quotient of g andf is g g͑x͒ Ί4 Ϫ x 2 ΂ ΃͑x͒ ϭ ϭ . f f ͑x͒ Ίx The domain off is ͓0, ϱ͒ and the domain of g is ͓Ϫ2, 2͔. The intersection of f g these domains is ͓0, 2͔. So, the domains of ΂ ΃ and ΂ ΃ are as follows. g f f g Domain of ΂ ΃ : ͓0, 2͒ Domain of ΂ ΃ : ͑0, 2͔ g f Note that the domain of ͑f͞g͒ includes x ϭ 0, but not x ϭ 2, because x ϭ 2 yields a zero in the denominator, whereas the domain of ͑g͞f͒ includes x ϭ 2, but not x ϭ 0, because x ϭ 0 yields a zero in the denominator. Now try Exercise 5(d). 333202_0108.qxd 12/7/05 8:43 AM Page 86 86 Chapter 1 Functions and Their Graphs Composition of Functions Another way of combining two functions is to form the composition of one with the other. For instance, if f ͑x͒ ϭ x2 and g͑x͒ ϭ x ϩ 1, the composition of f with g is f ͑g͑x͒͒ ϭ f ͑x ϩ 1͒ ϭ ͑x ϩ 1͒2. This composition is denoted as ͑ f Њ g͒ and reads as “f composed with g.” f g ˚ Definition of Composition of Two Functions The composition of the function f with the function g is g(x) x ͑ f Њ g͒͑x͒ ϭ f ͑g͑x͒͒. g f f(g(x)) Domain of g The domain of ͑ f Њ g͒ is the set of all x in the domain of g such that g͑x͒ is Domain of f in the domain of f. (See Figure 1.90.) FIGURE 1.90 Example 4 Composition of Functions Given f ͑x͒ ϭ x ϩ 2 and g͑x͒ ϭ 4 Ϫ x2, find the following. a.͑ f Њ g͒͑x͒ b.͑g Њ f ͒͑x͒ c. ͑g Њ f ͒͑Ϫ2͒ Solution a. The composition off with g is as follows. The following tables of values ͑ f Њ g͒͑x͒ ϭ f ͑g͑x͒͒ Definition of f Њ g help illustrate the composition ϭ ͑ Ϫ 2͒ ͑ ͒ ͑ f Њ g͒͑x͒ given in Example 4. f 4 x Definition of g x ϭ ͑4 Ϫ x 2͒ ϩ 2 Definition of f ͑x͒ x 012 3 ϭϪx 2 ϩ 6 Simplify. g͑x͒ 430Ϫ5 b. The composition of g withf is as follows. g͑x͒ 430Ϫ5 ͑g Њ f ͒͑x͒ ϭ g͑ f ͑x͒͒ Definition of g Њ f f ͑g͑x͒͒ 652Ϫ3 ϭ g͑x ϩ 2͒ Definition of f ͑x͒ ϭ 4 Ϫ ͑x ϩ 2͒2 Definition of g͑x͒ x 012 3 ϭ 4 Ϫ ͑x2 ϩ 4x ϩ 4͒ Expand. f ͑g͑x͒͒ 652Ϫ3 ϭϪx2 Ϫ 4x Simplify. Note that the first two tables can Note that, in this case, ͑ f Њ g͒͑x͒ ͑g Њ f ͒͑x͒. be combined (or “composed”) c. Using the result of part (b), you can write the following. to produce the values given in the third table. ͑g Њ f ͒͑Ϫ2͒ ϭϪ͑Ϫ2͒2 Ϫ 4͑Ϫ2͒ Substitute. ϭϪ4 ϩ 8 Simplify. ϭ 4 Simplify. Now try Exercise 31. 333202_0108.qxd 12/7/05 8:43 AM Page 87 Section 1.8 Combinations of Functions: Composite Functions 87 Technology Example 5 Finding the Domain of a Composite Function You can use a graphing utility Given f ͑x͒ ϭ x2 Ϫ 9 and g͑x͒ ϭ Ί9 Ϫ x2, find the composition ͑ f Њ g͒͑x͒. Then to determine the domain of a find the domain of ͑ f Њ g͒. composition of functions. For the composition in Example 5, enter Solution the function composition as ͑ f Њ g͒͑x͒ ϭ f ͑g͑x͒͒ 2 y ϭ ͑Ί9 Ϫ x2 ͒ Ϫ 9. ϭ f ͑Ί9 Ϫ x2 ͒ You should obtain the graph ϭ ͑Ί9 Ϫ x2 ͒2 Ϫ 9 shown below. Use the trace ϭ 9 Ϫ x2 Ϫ 9 feature to determine that the x-coordinates of points on the ϭϪx2 graph extend from Ϫ3 to 3. So, the From this, it might appear that the domain of the composition is the set of all real domain of ͑f Њ g͒͑x͒ is Ϫ3 ≤ x ≤ 3. numbers. This, however is not true. Because the domain of f is the set of all real 1 numbers and the domain of g is Ϫ3 ≤ x ≤ 3, the domain of ͑ f Њ g͒ is −5 5 Ϫ3 ≤ x ≤ 3. Now try Exercise 35. In Examples 4 and 5, you formed the composition of two given functions. In calculus, it is also important to be able to identify two functions that make up a −10 given composite function. For instance, the function h given by h͑x͒ ϭ ͑3x Ϫ 5͒3 is the composition of f with g, where f ͑x͒ ϭ x3 and g͑x͒ ϭ 3x Ϫ 5. That is, Activities 1. Given f ͑x͒ ϭ 3x2 ϩ 2 and g͑x͒ ϭ 2x, h͑x͒ ϭ ͑3x Ϫ 5͒3 ϭ ͓g͑x͔͒3 ϭ f͑g͑x͒͒. find f Њ g. Answer: ͑ f Њ g͒͑x͒ ϭ 12x2 ϩ 2 Basically, to “decompose” a composite function, look for an “inner” function and ͑ ͒ ϭ Ϫ 2. Given the functions an “outer” function.