If a relation is a function, how many y-values can an x-value have?
Domain: {5, 6, - 2} Range: { 3, 1, - 8} Is It a Function? Yes Is it one-to-one, onto, both, or neither? Both
Practice makes perfect
Domain is a set of all possible inputs, or x-values. Range is a set of all possible outputs, or y-values. Let's list the elements in the domain and range.
Domain:& { 5, 6, - 2}
Range:& { 3, 1, - 8}
For a relation to be a function, each x-value can only be paired with one y-value, but one y-value can be paired with multiple x-values. In a mapping, the x-values in the domain can only point to one y-value in the range.
In the given mapping, we can see that none of the x-values correspond to multiple y-values. Since all of the values in the domain only point to one value in the range, the relation is a function. In order to determine if a function is one-to-one, onto, both, or neither, let's review what each of these types of relationship means.
One-to-one: Each element of the domain is paired with exactly one unique element of the range. In other words, this type of function would pass a horizontal line test.
Onto: Every element in the range must be paired with at least one element of the domain.
Both: Each element of the domain is paired with exactly one element of the range and each element in the range is paired with exactly one element in the domain.
Observing the given mapping, no elements in the domain map to the same element in the range, and each element of the range corresponds to at least one element in the domain. Therefore, the function is both one-to-one and onto.
