Here is exactly how this duty looks on a graph v an x-extent the <-10, 10> and a y-extent that <-10, 10>:
First, an alert the x- and y-axes. Castle are drawn in red.
You are watching: Graph of y=1/x
The function, f(x) = 1 / x, is drawn in green.
Also, notification the slight flaw in graphing modern technology which is generally seen when illustration graphs that rational functions with computer systems or graphic calculators. At the bottom facility of the snapshot you will check out that the graph line appears to be heading toward the leaf of the diagram, but is cut quick of that. Actually, the true graph that the function continues downward past the sheet of the picture. As we will shortly see, this section of the graph holds what is termed an asymptote, and also computers, in addition to graphic calculators, regularly have a difficult time illustration functions close to asymptotes.
Notice that for this role a tiny positive input value yields a big positive output value. And notification that a huge positive input worth yields a tiny positive calculation value. Below is a picture showing that:
A complementary situation occurs for negative values. A small negative input will certainly output a big negative value, and a huge negative input will certainly output a small an adverse value. Right here is a photo showing this idea:
This makes complete sense if friend think around it because that a moment. Consider the intake x to have actually a big positive value, to speak one million (1,000,000). The calculation of f(x)=1/x would certainly be one millionth (1/1,000,000 or 0.000001).
This and also other representative examples are displayed in the following table:
|Input value, or x||Output value, or y|
|1,000,000||⇒||0.000001 (or 1/1,000,000)|
|0.000001 (or 1/1,000,000)||⇒||1,000,000|
|-1,000,000||⇒||-0.000001 (or -1/1,000,000)|
|-0.000001 (or -1/1,000,000)||⇒||-1,000,000|
Let us look in ~ this role as it leaves the graph at the top and also bottom. Girlfriend should notification that the green duty line approaches, but does not touch, the y-axis.
If you graphed the function on a collection of x, y axes that went increase to hopeful one million and also down to an adverse one million, the role line would certainly still not touch the y-axis, though it would get really close.
Just think around the x, y worths in the table above. At, say, an calculation value, or y value, of 1,000,000 the input would not be 0. And, the course, the input value, or x value, should be 0 because that the graph come touch the y-axis.
This kind of behavior around the y-axis is dubbed asymptotic behavior. And, in this case, the y-axis would certainly be called a upright asymptote that the function. The is, the role approaches the y-axis ever before closer and also closer, however never touches it.
Notice the the x-axis features as a horizontal asymptote because that this function. That is, together the function line stretches the end to the left or right it gets closer and closer come the x-axis, yet it never ever touches it.
So, because that the role f(x)=1/x the y-axis is a vertical asymptote, and also the x-axis is a horizontal asymptote. In the complying with diagram of this role the asymptotes are drawn as white lines.
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The duty f(x)=1/x is great starting point from i m sorry to build an understanding of rational functions in general. It is a polynomial divided by a polynomial, back both room quite an easy polynomials. Be sure that you know the concept of one asymptote, specifically a upright asymptote, and also then walk on come the various other rational function information.