Question

Definition 5.48. Let f,g:X + Y be functions and assume that Y is a set in which the following operations make sense. Then the following are also functions: 1. f + g defined by (f +g)(x) = f(x) + g(x) for all x E X 2. f - g defined by (f – g)(x) = f(x) – g(x) for all x € X 3. f.g defined by (fºg)(x) = f(x) · g(x) for all x E X f(x) = "147 for all x E X such that J(W) EY. g(x) Notice that each of parts (1) through (4) of the preceding definition defines a (possibly different) function whose domain is a subset of X. Question 5.49. Why is the phrase "possibly different” in the preceding definition? Can it ever be the case that any of the functions (f +g), (f – 9), (fºg) or () are the same as either f or g? 4. I defined g(x) g Question 5.50. What are the possibilities for the codomains of the functions in Definition 5.48? Can any of them be anything other than Y? Example 5.51. Suppose f and g are functions with the same domain and codomain. What is f +g? Solution. Since f and g are functions, each can be represented by a set of ordered pairs. Then we can write f + g as f+9 = {(x,y + y')| (x, y) e f and (x, y') Eg}. Exercise 5.52. Express each of the other three new functions in Definition 5.48 as a set of ordered pairs.

Answer 1

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