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How To Graph The Inverse Of A Function

Asked by mary jones, student, Dan River High School on September 27, 1997:
How do you graph an inverse function once you have solved it?
It is actually easier to graph the inverse of a function than it is to solve for it.

First let's think about what it means to graph a function. Suppose that we wanted to graph a function f(x). To do this, we would substitute numerical values for x and plot those ordered pairs (x,y) for which y=f(x). For instance, if  (IMAGE) , we might try plotting the points (0,0), (1,1), (2,4), (3,9), etc.

Now how would we plot the inverse of a function? If g is the inverse function of f, then to graph g we'd plot the points (x,y) where y=g(x). This condition is the same as the condition x=f(y), so the graph of the inverse consists of points (f(y),y).

These are the same as the points on the graph of f but with the order of the coordinates interchanged: instead of plotting (x,f(x)) for various numerical values of x, we plot (f(y),y) for various numerical values of y (which is the same thing as plotting (f(x),x) for various numerical values of x). For example, some of the points on the graph of the inverse of  (IMAGE) are (0,0), (1,1), (4,2), and (9,3). Geometrically, we have just reflected the graph of the function f(x) through the line y=x to get the graph of the inverse of f(x).

This method of graphing the "inverse" of a function always works, even when the function doesn't have an inverse. If the function doesn't have an inverse, it is because there are two distinct values a and b which we can assign to x to get the same value for f(x). If we examine our function  (IMAGE) we will note that f(2) = f(-2) = 4. The corresponding points on the graph of our "inverse function" are (4,2) and (4,-2). Thus the graph which we constructed in this method is not really the graph of a function, since the value of the inverse of f(x) is not well defined at 4 (it could either be 2 or -2).

Even though this approach will not always give us the graph of a function, it will whenever the inverse of f(x) exists.

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