Question

7. We list several pairs of functions f and g. For each pair, please do the following: Determine which of go f and fog is defined, and find the resulting function(s) in case if they are defined. In case both are defined, determine whether or not go f = fog. (a) f = {(1,2), (2,3), (3, 4)} and g = {(2,1),(3,1),(4,1)). (b) f = {(1,4), (2, 2), (3, 3), (4,1)} and g = {(1, 1), (2, 1), (3, 4),(4,4)}. (c) f(x)=x2-1 and g(x) = x2 +1 (both for all ze Z). (a) f(x) = x + 3 and g(x) = x - 7 (both for all r e Z). (e) f(0) = 1 - 2 and g(x) = 2 - 2 (both for all x E Q). 8. Let f: A + A and 9: A + A be functions such that go f = f. Must we have g = idA? Prove your answer.

Answer 1

fog + gof fog) = {(1,2), (3,2), (4,2) fog + gef (916) 64,1), (2,1), (3,1) gof = {(3,4), (3,1), (3,4), (4,1)) fog = {(1,4), (2,4), (3,1), (4,177 (fog)(2) = f(2®+1) = (272 (gof) (1) = 9(2+1) = (2-1)+ IP = 24-27 + 2 i fogt gof (goff²) = 9(x+3) = (x+37 7 = 2-4 (fog) (7) = f (x-7) = (2-7) +3 = x-4 fog &=g of . +222

= 1+2 (fog)(x) = f(2-x) = 1-(2-2) g•f) (a) = 9(1-2) = 2-(1-2) * 9 * *f fiA A : AA g.f = f f g = ida Take A = {1,2,3} f(2)=1 #2EA. g1Q =/ 1, 2= 1 g+ IdA but gof)(x) = f(x) VREA