8.1 FUNCTIONS AND MAPPINGS 265
(c) Prove that iff and g are linear mappings of R into R, that is f(x) = Mx + B
and g(x) = Nx + C, then f 0 g and g 0 f are both linear mappings with slope MN.
(d) [Continuation of (c)] Determine conditions on M, N, B, and C such that
f 0 g = g 0 f. Give a specific example of linear mappings f and g such that
f os=sof.
(e) Consider two functions f and g, where f # (25 and g # 0. Find a condition
involving domains and ranges of these functions that guarantees that g 0 f = 0.
- Let f: A -+ B be any mapping, let I, and I, be the identity mappings on A and
B, respectively. Prove:
(a) f .IA= Ie0 f = f
(6) Iff is injective, then f 0 f = I,.
*(c) If X E A, then f^0 i, = fl,, where i, is the inclusion mapping of X into A.
- (a) Complete the proof of the "if" part of Theorem 4.
(6) Prove that if f: A -+ B and g: B -+ C are injections, the g 0 f: A -, C is an
injection.
(c) Prove that iff: A -+ B and g: B + C are mappings such that g 0 f: A -+ C is
an injection, then f is an injection.
(d) In each of the following two cases, give examples of two mappings f: A -, B
and g: B -+ C such that:
(i) g^0 f: A -+ C is an injection, but g is not an injection (Note: What must be
true off in any such example?)
(ii) f is injective, but g 0 f is not injective (Note: What must be true of g in any
such example?)
*(e) Suppose that f: A -+ B and g: C -+ D are mappings and assume rng f r C.
Prove that dom (g 0 f) = A and rng (g 0 f) E D.
- Let f and g be mappings of R into R. Define new mappings f v g and f A g from
R into R by (f v g)(x) = max { f (x), g(x)) and (f A g)(x) = min { f (x), g(x)) for each
x E R. Give examples to show that f v g and f A g can fail to be one to one, even
when f and g are both one to one.
- Let f: A -+ B and g: C -+ D be mappings:
(a) Prove that if A n C = 0, then the relation f u g (the set theoretic union of
f and g, regarded as sets of ordered pairs) is a mapping of A u C into B u D.
(6) Prove that if A n C = B n D = 0 and iff and g are injective, then f u g is
injective.
(c) Prove that if A n C = @, then (f u g)/, = f and (f u g)/, = g.
- Let f: A -+ B be a mapping. A mapping g: C -+ B is said to be an extension of
f if and only iff E g.
*(a) Prove that if g: C -+ B is an extension off: A -+ B, then A c C.
(6) Prove that if f: A -+ B and g: C -+ D are mappings with A n C = @, then
f u g: A u C -+ B u D is an extension of both f and g.
(c) Prove that if g: C -, B is an extension off: A + B, then g/, = f.
(d) Find an extension of the mapping f(x) = (x2 - 25)/(x + 5) of R - {- 5) into
R, whose domain is R and that is continuous on R.
(e) Show that the mapping g: C -+ C defined by g(z) = eX(cos y + i sin y), where
z = x + yi, is an extension of the mapping f: R -+ C given by f(x) = 8.
