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Computing k 1 is left as an exercise 1 1 y k x as

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Unformatted text preview: eflect across y = x −− − − − − − − − − − − − − −→ switch x and y coordinates (4, −2) y = f −1 ( x ) ? We see that the line x = 4 intersects the graph of the supposed inverse twice - meaning the graph fails the Vertical Line Test, Theorem 1.1, and as such, does not represent y as a function of x. The vertical line x = 4 on the graph on the right corresponds to the horizontal line y = 4 on the graph of y = f (x). The fact that the horizontal line y = 4 intersects the graph of f twice means two different inputs, namely x = −2 and x = 2, are matched with the same output, 4, which is the cause of all of the trouble. In general, for a function to have an inverse, different inputs must go to different outputs, or else we will run into the same problem we did with f (x) = x2 . We give this property a name. Definition 5.3. A function f is said to be one-to-one if f matches different inputs to different outputs. Equivalently, f is one-to-one if and only if whenever f (c) = f (d), then c = d. Graphically, we detect one-to-one functions using the test below. Theorem 5.4....
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