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f(x) = 1 / x
This is probably the simplest of rational functions:
Here is how this function looks on a graph with an x-extent of [-10, 10] and a y- extent of [-10, 10]:
First, notice the x- and y-axes. They are drawn in red.
The function, f(x) = 1 / x, is drawn in green.
Also, notice the slight flaw in graphing technology which is usually seen when drawing graphs of rational functions with computers or graphic calculators. At the bottom center of the picture you will see that the graph line appears to be heading toward the edge of the diagram, but is cut short of that. Actually, the true graph of the function continues downward past the edge of the picture. As we
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will shortly see, this section of the graph holds what is termed an asymptote, and computers, along with graphic calculators, often have a difficult time drawing functions near asymptotes.
Notice that for this function a small positive input value yields a large positive output value. And notice that a large positive input value yields a small positive output value. Here is a picture showing that:
A complementary situation occurs for negative values. A small negative input will output a large negative value, and a large negative input will output a small negative value. Here is a picture showing this idea:
This makes complete sense if you think about it for a moment. Consider a large
The function f(x) = 1/x is an excellent starting point from which to build an understanding of rational functions in general. It is a polynomial divided by a polynomial, although both are quite simple polynomials. Be sure that you understand the concept of an asymptote, especially a vertical asymptote, and then go on to the other rational function information.
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