How do you determine whether a set of ordered pairs is a function?

Answer 1

If f is a set of ordered pairs, then f is a function from X to Y if and only if all three of the following hold:

#1: ∀(x, y)∈f, x∈X ∧ y∈Y
#2: ∀x∈X, ∃y s.t. (x, y)∈f
#3: ∀(a, b)∈f ∀(c, d)∈f, (a=c ⇒ b=d)

In other words, you have to check that:

#1: all the ordered pairs map an element in the domain to an element in the codomain
#2: every element in the domain is mapped to at least one element in the codomain
#3: No element in the domain is mapped to more than one element in the codomain.

Note that if you're just given the set of ordered pairs and are not told the purported domain and codomain of the function, you generally assume that the domain is just {x: ∃y s.t. (x, y)∈f} and the codomain is any convenient superset of the range, so that #1 and #2 are automatic. Thus you would only have to check #3 in that case -- that is, there are no pairs of ordered pairs that have the same first element and different second elements.

Answer 2

A function is a set of ordered pairs in which no two pairs with the same first element have a different second element.

If you think about it, it makes sense. Consider f(x) = x + 1. The pairs (x, f(x)) look like (1,2), (2,3), (3,4), (4,5), etc. If you also had (1,3) in the set, then there'd be two different answers for 1+1. The same applies to any function. There can only be one result for a given input value.

Answer 3

A function is determined by two collections A and B and an assignment of a unique element of B to each element of A.

Ordered Pair (A,B) the same as (x,y).

Answer 4

A function is a collection of ordered pairs that has precisely a million x-coordinate for a million y-coordinate. the 1st because of the fact the -a million is repeated two times interior the variety, it fairly is not a function. comparable with type 2: 10 is repeated two times interior the area.

Answer 5

you know if a set of ordered pairs is a function is for every x value, there is only one distict y value.

for example, if (2,3) and (2,5) existed at the same time on the graph, the graph would not be a function.

Answer 6

the easiest way is looking if the domain integers (X-value) are all different. Range (y) dont matter.

(1,0) (2,0) is a function

(0,1) (0,2) is not