# How do you know whether a relation is a function and what the domain and range is for (1,-3), (6,-2), (9,-1), (1,3)?

Mar 25, 2015

This relation is not a function.
The domain of this relation is $\left\{1 , 6 , 9\right\}$
The range of this relation is $\left\{- 3 , - 2 , - 1 , 3\right\}$

#### Explanation:

A relation is a function if no two ordered pair in the relation have the same first element and different second elements.
(That is; If two different pair have the same first element then the relation is not a function.)

In $R = \left\{\begin{matrix}1 & - 3 \\ 6 & - 2 \\ 9 & - 1 \\ 1 & 3\end{matrix}\right\}$

the pairs $\left(1 , - 3\right)$ and $\left(1 , 3\right)$ cause this relation to not be a function.

The domain if the set of all first elements.
"Domain"(R)={x| (EEy) [(x,y) in R}

So the domain of your relation is $\left\{1 , 6 , 9\right\}$

The range is the set of second elements.
"Range"(R)={y| (EEx) [(x,y) in R}

So the range of your relation is $\left\{- 3 , - 2 , - 1 , 3\right\}$

Note
(Not all introductory level classes include domains and ranges for arbitrary relations. Some only mention domain and range for functions.)

Sep 17, 2017

The given points DO NOT represent a function.

Domain$= \left\{1 , 6 , 9\right\}$

Range $= \left\{- 3 , - 2 , - 1 , 3\right\}$

#### Explanation:

A relation is a function if, for EACH $x$ value there is only ONE possible $y$ value.

As soon as there is a choice for $y$, it is not a function.

So (3,1); (2,1); (1,1); (0,1) IS a function because there is no choice for the $y$ values.

It does not matter than several $x$-values all have the same $y$.

(1,3);(1,4); (1,5); (1,6) is NOT a function, because for the $x$ value $x = 1$, the $y$ values can be $3 \mathmr{and} 4 \mathmr{and} 5 \mathmr{and} 6$

In the given ordered pairs, if $x = 1$, then $y = 3 \mathmr{and} - 3$
This is therefore NOT a function.

The Domain is the set of all the $x$-values.
The Range is the set of all the $y$ values.

For the given ordered pairs:

Domain$= \left\{1 , 6 , 9\right\}$

Range $= \left\{- 3 , - 2 , - 1 , 3\right\}$