Let f : A → B and g : B → C and define g ◦ f : A → C by (g ◦ f)(x) := g(f (x)).
a) Show that if f, g are 1–1 (respectively, onto), then g ◦ f is 1–1 (respectively, onto).
b) Prove that if f is 1–1 from A into B and B0 := {y : y = f (x) for some x ∈ A}, then f−1 is 1–1 from B0 onto A.
c) Suppose that g is 1–1 from B onto C. Prove that f is 1–1 on A (respectively, onto B) if and only if g ◦ f is 1–1 on A (respectively, onto C).
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