Many functions have so-called asymptotes - straight lines that the graph approaches with increasing distance to the origin
of the coordinate system.
Especially vertical asymptotes are often immediately obvious when looking at the function rule: any values for `x` that would require you to divide by would result in such an asymptote.
A horizontal asymptote is seen if the function value approaches a certain `y` -value when input values get either very large or very small (large negative numbers).
The function with rule is the prototype of a function with asymptotes. The graph of this function is shown here. This graph has:
a horizontal asymptote given by the line , because the larger or smaller (more negative) values of become, the closer the function value gets to 0;
a vertical asymptote that is given by the line , because there is no function value for this value of `x` (you cannot divide by ), and the closer the line gets to the origin (to ) the larger the function values get (in positive or negative direction).
The domain of
And the range is .
The sign indicates the combined set of all numbers in the two intervals.
When you choose a proper window to plot the graph of funtion , then all its characteristics will be visible. These characteristics are:
the points of intersection with the axes, which are the zeros (or roots) of the function and the -intercept;
the vertices, the points where the function has (local) maxima and minima, which you can find using your graphing calculator.