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)`?

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={(1,-3), (6,-2), (9, -1), (1,3)}`

the pairs `(1, -3)` and `(1, 3)` 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 `{1, 6, 9}`

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

So the range of your relation is `{-3, -2, -1, 3}`

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

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 or 4 or 5 or 6`

In the given ordered pairs, if `x=1`, then `y=3 or -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`= {1,6,9}`

Range `={-3,-2,-1,3}`