Functions and graphs > Characteristics
12345Characteristics

## Theory

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 $0$ 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 $f$ with rule $f\left(x\right)=\frac{1}{x}$ 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 $y=0$, because the larger or smaller (more negative) values of $x$ become, the closer the function value gets to 0;

• a vertical asymptote that is given by the line $x=0$, because there is no function value for this value of $x$ (you cannot divide by $0$), and the closer the line gets to the origin (to $0$) the larger the function values get (in positive or negative direction).

The domain of $f$ is ${\text{D}}_{f}=⟨←,0⟩\cup \left(0,\to ⟩$.
And the range is ${\text{B}}_{f}=⟨←,0⟩\cup \left(0,\to ⟩$.
The $\cup$ sign indicates the combined set of all numbers in the two intervals.

When you choose a proper window to plot the graph of funtion $f$, 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 $y$ -intercept;

• the asymptotes;

• the vertices, the points where the function has (local) maxima and minima, which you can find using your graphing calculator.