Functions and graphs

Functions: Introduction to functions

Functions and graphs

We often work with $x,y$ -plane, whose elements are the pairs $\rv{x,y}$ of real numbers $x$ and $y$ . In this plane we can visualize functions.

Graph

A graph is a set of points of the plane.

A function $f$ can be made visible by drawing a graph: each value of $x$ from the domain of $f$ corresponds to a unique value for $y$ , being $f(x)$ . The set of points $\rv{x,y}$ thus obtained, is the graph of the function $f$ .

In other words, the graph of $f$ consists of all $\rv{x,f(x)}$ for $x$ in the domain of $f$ .

It depends on the domain of $f$ .

A function determines a graph. But what about the converse: when you are given a set $X$ of points in the real plane, is there a function whose graph coincides with $X$ ? The answer is not always affirmative:

Graphs that are derived from a function

To each value of $x$ in the domain a function, the function assigns exactly one value of $y$ , and to each value of $x$ outside the domain, the function assigns no value of $y$ . Thus the graph of a function has the property that each vertical line meets the graph in at most one point.

The converse is also true: if $X$ is a graph such that every vertical line of the plane meets $X$ in at most one point, then $X$ is the graph of a function.

A circle is not the graph of a function. When we take a look at the circle in the $x,y$ -plane centered at the origin, we quickly see that we do not find one single function to describe it. After all, there should only be one value of $y$ for each value of $x$ in the domain. Now consider the $x$ axis, $x=0$ . It meets the above-mentioned circle in two point: with $y$ -coordinates $y=1$ and $y=-1$ . This is in conflict with the rule. So there is no function whose graph is this circle.

If $X$ is the graph of a function $f$ , then the domain of $f$ is the set of points $x$ such that the vertical line through $\rv{x,0}$ meets $X$ (in a single point).

Consider the linear function $f(x)=-x-7$ . The graph of this function is a straight line. Please draw the unique line that is the graph of $f$ below.

The line through $\rv{0,-7}$ and $\rv{2,-9}$

A line is determined by two of its points. We calculate the points of the graph with $x$ -coordinates $0$ and $2$ , respectively:

$x=0$ gives $f(0)=-7$
$x=2$ gives $f(2)=2\cdot -1 -7=-9$

Thus, the two points $\rv{0,-7}$ and $\rv{2,-9}$ belong to the graph of $f$ . The line through these points is the graph of $f(x)=-x-7$ .

New example