A relation is any set of points which connect two variables.
A relation is often expressed in the form of an equation connecting the variables $x$ and $y$. In this case the relation is a set of points $(x,y)$ in the Cartesian plane. This plane is separated into four quadrants according to the signs of $x$ and $y$. For example, $y=x+1$ and $x=y^2$ are the equations of two relations. Each equation generates a set of ordered pairs, which we can graph:
FunctionsA function, sometimes called a mapping, is a relation in which now two different ordered pairs have the same $x$-coordinate or first component.
We can see from the above definition that a function is a special type of relation.
Every function is a relation, but not every relation is a function.
Testing for FunctionsAlgebraic Test: If a relation is given as an equation, and the substitution of any value for $x$ results in one and only one value of $y$, then the relation is a function.
$y=x+1$ is a function, as for any value of $x$ there is only one corresponding value of $y$.
- $x=-2 \rightarrow y=-1$
- $x=-1 \rightarrow y=0$
- $x=0 \rightarrow y=1$
- $x=1 \rightarrow y=2$
- $x=2 \rightarrow y=3$
- $x=4 \rightarrow y=2$
- $x=4 \rightarrow y=-2$
- $x=1 \rightarrow y=1$
- $x=1 \rightarrow y=-1$
- is a function if each line cuts the graph no more than once
- is not a function if at least one line cuts the graph more than once