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Unformatted text preview: eorem 5.5. Equivalent Conditions for Invertibility: Suppose f is a function. The following statements are equivalent. • f is invertible. • f is one-to-one. • The graph of f passes the Horizontal Line Test. We put this result to work in the next example. Example 5.2.1. Determine if the following functions are one-to-one in two ways: (a) analytically using Definition 5.3 and (b) graphically using the Horizontal Line Test. 1 − 2x 5 2x 2. g (x) = 1−x 3. h(x) = x2 − 2x + 4 1. f (x) = 4. F = {(−1, 1), (0, 2), (2, 1)} Solution. 1. (a) To determine if f is one-to-one analytically, we assume f (c) = f (d) and attempt to deduce that c = d. f (c) 1 − 2c 5 1 − 2c −2c c = f (d) 1 − 2d = 5 = 1 − 2d = −2d =d Hence, f is one-to-one. (b) To check if f is one-to-one graphically, we look to see if the graph of y = f (x) passes the Horizontal Line Test. We have that f is a non-constant linear function, which means its graph is a non-horizontal line. Thus the graph of f passes the Horizontal Line Test as seen below....
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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