questions:

(1) Do there exist functions f and g defined on R such that f(x) + 9(y) = ry for all real numbers a and y? Explain. (2) Your friend Susan has become interested in functions f: R -+ R which preserve both the operation of addition and the operation of multiplication; that is, functions f which satisfy f(a + y) = f(x) + f(y) (3.1) and f(ry) = 1(2)f(y) (3.2) for all r, y E R. Naturally she started her investigation by looking at some examples. The trouble is that she was able to find only two very simple examples: f(x) = 0 for all a and f(x) = x for all z. After expending considerable effort she was unable to find additional examples. She now conjectures that there are no other functions satisfying (3.1) and (3.2). Write Susan a letter explaining why she is correct. Hint. You may choose to pursue the following line of argument. Assume that f is a function (not identically zero) which satisfies (3.1) and (3.2) above. (a) Show that f(0) = 0. [In (3.1) let y = 0.] (b) Show that if a # 0 and a = ab, then b = 1. c) Show that f(1) = 1. How do we know that there exists a number e such that f(c) # 07 Let a = c and y = 1 in (3.2).] (d) Show that f(n) = n for every natural number n. (e) Show that f(-n) = -n for every natural number n. [Let a = n and y = -n in (3.1). Use (d).] (f) Show that f(1) = 1 for every natural number n. [Let a = n and y = 1 in (3.2).] (g) Show that /(r) = r for every rational number r. [If r 2 0 writer = m where m and n are natural numbers; then use (3.2), (d), and (e). Next consider the case r r or f(x) > and f(x)