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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 f(x) = x^2, 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 f(x)=x^2 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 f(x) = x^2 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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