Question

(d) Translate the following statement into predicate logic: “Every function f :R → R can be written as the sum of an even function and an odd function.” You can use the notation fi + f2 to represent the sum of functions fı and f2, and the notation f1 = f2 to represent the fact that fi and f2 are equal. 2n izo (e) Let n € N, and 20, 21, ..., Q2n E R. Let f: R + R be defined as f(x) = x aix’. Find functions fi: R → R and f2: R → R such that the following three properties hold: • fi is even, • f2 is odd, • f1 + f2 = f. Justify your responses, but no formal proof is necessary. Simplify fi and f2 where possible. (f) Let g: R +R be defined as g(x) = 24 instead? Find functions 91 : R + R and 92: R + R such that the following three properties hold: • 91 is even, • g2 is odd, • 91 +92 = g. Justify your responses, but no formal proof is necessary. Simplify 91 and 92 where possible. Hint: g(1) = 2 and g(-1) = . From this and the definition of even/odd functions, first determine 91(1), 92(1), 91(-1), and 92(1).

Answer 1

(d) for every function f. R R fo: h "R, there exist fR IR and R Suchohat t, is even even and and fo is fo is add add (e) ang ang f= fit fa. NEIN, ao, a - an IR f(x) = let Debine fi: R IR f (2) = 20 1994 and fa: R R f (x) = a(zi-D ALL L=1 agita Zaai (- = I asi ( = f(x) So fi is even. • fa (-x) = 1 a cai-D (-2) So fg B odd 51462 = 2 2 fitfg = 2 2 y 24 Žais! g. RR g(x) = 2 taking gi - 1 (g (-2) + g(x)) = à 2 x + 2 ) 92 a ž 18 Cox) - g (x)) = 2 2 2 2 ² • and gi tg2² 1 (2² +2²) + Can - 2 2) = 2 • g, (-x) = & Cat & +22) = 9, (x) hence g, is even • Ja (-x) = (2-2) = -92 (8) 2 o odd