1-1 functions, Inverse & Trig Functions, Intro to Limits

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Prove that if 0 < < /2, then cos < sin / < 1. A function f is called one-to-one (1-1) if it never takes the same value twice. That is, if , then f(x1) f(x2) Prove that f(x) = with domain d(f) = [1, ) is 1-1. Note bc < ba < arc ba sin < ba < thus sin < so. So arc ba < tan so < tan = Suppose f( ) = f( ) for some x1, x2 d(f) 2 = 0 x1 + x2 + x1x2 x1 - x2 + x1x2(x2 - x1) = 0 (x1 - x2)(1 - x1x2) = 0. If x1x2 = 1, then since x1x2 >= 1 (d(f) = [1, ], x1 = x2 = 1. In both cases x1 = x2, so f is one-to-one. If we consider g(x) = with domain d(g) = [0, ), then g(x) is not 1-1 (eg. g(2) = g(

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