First consider a simple example f(x) = 3x + 2. Graph the following functions and determine whether or not they have inverses. The following definition is equivalent, and it is the one most commonly given for one-to-one.Īlternate Definition: A function f is one-to-one if, for every a and b in its domain, The property of having an inverse is very important in mathematics, and it has a name.ĭefinition: A function f is one-to-one if and only if f has an inverse. If no horizontal line intersects the graph of f more than once, then f does have an inverse. If any horizontal line intersects the graph of f more than once, then f does not have an inverse. We can tell before we reflect the graph whether or not any vertical line will intersect more than once by lookingĪt how horizontal lines intersect the original graph! Horizontal Line Test Line y = x, the result is the graph of a function (passes the vertical line test). This generalizes as follows: A function f has an inverse if and only if when its graph is reflected about the Note that the reflected graph does not pass the vertical line test, The graph of f and its reflection about y = x are drawn below. If f had an inverse, then its graph would be the reflection of Look at the same problem in terms of graphs. Therefore, there is no function that is the inverse of f. ![]() On the other hand, since f(-2) = 4, the inverse of f would have to take 4 to -2. If f had an inverse, then the fact that f(2) = 4 would imply that the Some functions do not have inverse functions. Return to Contents Existence of an Inverse
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