Inverses Inverses
Example: The inverse of the relation (2,3), (4,5), (2,6), (4,6) is (3,2), (5,4), (6,2), (6,4) Generally we switch the roles of x and y to find the inverse. For functions, we follow the steps below to find the inverse:
Example Find the inverse of y = 2x + 1
Solution
Notice that the original function took x, multiplied by 2 and added 1, while the inverse function took x, subtracted 1 and divided by 2. The inverse function does the reverse of the original function in reverse order.
Exercises Find the inverse of
Graphing Inverses To graph an inverse we imaging folding the paper across the y = x line and copy where the ink smeared in the other side.
One to One Functions A function y = f(x) is called one to one if for every y value there is only one x value with y = f(x). That is, each y value comes from a unique x value. Example
The Horizontal Line Test If the graph of a function is such that every horizontal line passes through the graph at at most one point then the function is 1-1. The graph below is the graph of a 1 -1 function since every horizontal line crosses the graph at most once.
However, the graph below is not the graph of a 1 -1 function, since there is a horizontal line that crosses the graph more than once.
An application of this is if we want a computer to find an inverse function, then we first have the computer check to see if the function is one to one, then have it proceed to find the inverse.
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