# Relations and Functions

# Relations

A relation is any set of points that connect two variables.

A relation is often expressed as an equation connecting the variables $x$ and $y$. 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:

$\cdots,(-2,-1),(-1,0),(0,1),(1,2),(2,3),\cdots$

$\cdots,(4,2),(1,1),(0,0),(1,-1),(4,-2),\cdots$

# Functions

A function, sometimes called a mapping, is a relation in which two different ordered pairs now 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 Functions

**Algebraic 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.

For example,

$y=x+1$ is a function, as for any value of $x$ there is only one corresponding value of $y$.

- $x=-2 \rightarrow y=-1$
- $x=-1 \rightarrow y=0$
- $x=0 \rightarrow y=1$
- $x=1 \rightarrow y=2$
- $x=2 \rightarrow y=3$

$x=y^2$ is not a function since if $x=4$ then $y=\pm 2$. That is because one $x$ value corresponds to two $y$ values.

- $x=4 \rightarrow y=2$
- $x=4 \rightarrow y=-2$
- $x=1 \rightarrow y=1$
- $x=1 \rightarrow y=-1$

**Vertical Line Test or Geometric Test:** If we draw all possible vertical lines on the graph of a relation, the relation:

- 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

### Example 1

Is the set of ordered pairs ${(1,4),(2,5),(3,6),(4,7)}$ a function?

${(1,4),(2,5),(3,6),(4,7)}\rightarrow$ a function as for any value of $x$ there is only one corresponding value of $y$.

### Example 2

Is the set of ordered pairs ${(1,3),(2,2),(3,4),(4,2)}$ a function?

${(2,2),(4,2)}\rightarrow$ not a function as one $x$ value is corresponding two $y$ values.

### Example 3

Is the set of ordered pairs ${(0,0),(1,1),(2,2),(3,3)}$ a function?

${(0,0),(1,1),(2,2),(3,3)}\rightarrow$ a function as for any value of $x$ there is only one corresponding value of $y$.

### Example 4

Is the set of ordered pairs ${(0,0),(1,0),(2,0),(3,0)}$ a function?

${(0,0),(1,0),(2,0),(3,0)}\rightarrow$ a function as for any value of $x$ there is only one corresponding value of $y$.

### Example 5

Is the set of ordered pairs ${(1,0),(1,1),(1,2),(1,3)}$ a function?

${(1,0),(1,1),(1,2),(1,3)}\rightarrow$ not a function as one $x$ value is corresponding two $y$ values.

### Example 6

Is the graph a function?

The graph is a function as each vertical line cuts the graph only once.

### Example 7

Is the graph a function?

The graph is not a function, as each vertical line cuts the graph more than once.

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