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.
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