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Determine whether the function is one-to-one. f(x)=(x-5)^3
Horizontal Line Test • If some horizontal line intersects the graph of the function more than once, then the function is not one-to-one. • If no horizontal line intersects the graph of the function more than once, then the function is one-to-one. What are One-To-One Functions? Algebraic Test Definition 1. A function f is said to be one-to-one (or injective) if f(x1) = f(x2) implies x1 = x2. Lemma 2. The function f is one-to-one if and only if 8x1, 8x2, x1 6= x2 implies f(x1) 6= f(x2). Examples and Counter-Examples Examples 3. • f(x) = 3x − 5 is 1-to-1. • f(x) = x2 is not 1-to-1. • f(x) = x3 is 1-to-1. • f(x) = 1 x is 1-to-1. • f(x) = xn − x, n > 0, is not 1-to-1. Proof. • f(x1) = f(x2) ) 3x1 − 5 = 3x2 − 5 ) x1 = x2. In general, f(x) = ax − b, a 6= 0, is 1-to-1. • f(1) = (1)2 = 1 = (−1)2 = f(−1). In general, f(x) = xn, n even, is not 1-to-1. • f(x1) = f(x2) ) x31 = x32 ) x1 = x2. In general, f(x) = xn, n odd, is 1-to-1. • f(x1) = f(x2) ) 1 x1 = 1 x2 ) x1 = x2. In general, f(x) = x−n, n odd, is 1-to-1. • f(0) = 0n − 0 = 0 = (1)n − 1 = f(1). In general, 1-to-1 of f and g does not always imply 1-to-1 of f + g. Properties of One-To-One Functions Properties Properties If f and g are one-to-one, then f g is one-to-one. Proof. f g(x1) = f g(x2) ) f(g(x1)) = f(g(x2)) ) g(x1) = g(x2) ) x1 = x2. Examples 4. • f(x) = 3x3 − 5 is one-to-one, since f = g u where g(u) = 3u − 5 and u(x) = x3 are one-to-one. • f(x) = (3x − 5)3 is one-to-one, since f = g u where g(u) = u3 and u(x) = 3x − 5 are one-to-one. • f(x) = 1 3x3−5 is one-to-one, since f = g u where g(u) = 1 u and u(x) = 3x3 − 5 are one-to-one. 2