The graph of a function.

Let $ f$ be a real function of a real variable $ x$ , defined over a domain $ \mathcal{D}$ in $ \mathbb{R}$ . The set of points in the plane whose coordinates are $ (x, f(x))$ , for $ x \in \mathcal{D}$ is called the graph of $ f$ .

Example 2.5.1   In figure [*] we display the graphs of the absolute value function ( $ \forall x \in \mathbb{R}, \; f(x)=\vert x\vert \;
$ ) and of the floor function (=integer part function: $ \forall x \in
\mathbb{R}, \; f(x)=[x] \; $ ).

Figure 2: The graphs of two well-known functions.
\begin{figure}\mbox{
\subfigure[absolute value]{\epsfig{file=AbsoluteValue.eps,...
...
\subfigure[floor]{\epsfig{file=IntegerPart.eps,height=4.5cm}}
}\end{figure}

In figure [*] we display the graphs of two other well-known functions; these graphs are a parabola and an hyperbola, respectively.

Figure 3: The graphs of two other well-known functions.
\begin{figure}\mbox{
\subfigure[$f(x)=x^2$]{\epsfig{file=parabola1.eps,height=4...
...figure[$f(x)=\frac 1x$]{\epsfig{file=hyperbola.eps,height=4cm}}
}\end{figure}

The curves in figure 4 are not graphs of functions. Dear reader, why?

Figure 4: Curves which are not graphs of functions.
\begin{figure}\mbox{
\subfigure[ellipse]{\epsfig{file=ellipse.eps,height=3.5cm}...
...subfigure[another curve]{\epsfig{file=aCurve.eps,height=3.5cm}}
}\end{figure}

We will make great use of graphs. On the one hand, they provide a good visual intuition of the phenomena described by the fuctions. On the other hand, applied scientists are often able to draw a graph representing the phenomenon that they study, even when the actual function is still unknown. From the study of the graph they can get many useful conclusions.

Noah Dana-Picard 2007-12-28