Reference no: EM131019983
1. Let I be an interval of the real line, and let f be a real-valued function with I ⊆ Dom( f). We say that f is increasing on I iff for all x, y ∈ I, if x < y, then f (x) < f(y). We say that f is decreasing on I iff for all x, y ∈ I, if x < y, then f (x) > f (y). Prove that
(a) g is decreasing on (-∞, 3), where g(x) = (x + 1) / (x - 3)
(b) h is decreasing on I, where h(x) = - f (x) and f is increasing on I.
(c) f is increasing on I, where f = g o h, and g and h are increasing on I.
2. Let f: A → B and g: C → D. Define
f x g = {((a, c), (b, d)): (a, b) ∈ f and (c, d) ∈ g}.
Prove that fxg:AxC → BxD. For (a, c) ∈ A x C, find an explicit expression for (f x g)(a, c).
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