How to Check Whether a Relation Is a Function

Quick answer: A relation from a stated domain is a function when every domain element has exactly one output. Different inputs may share an output; one input cannot have two distinct outputs.

Worked example

Is {(1, 3), (2, 3), (3, 5)} a function on {1, 2, 3}?

1. Input 1 has output 3.
2. Input 2 also has output 3; sharing is allowed.
3. Input 3 has output 5.
4. Every domain element has one output, so this is a function.

Check the result

Its range is {3, 5}; there are three inputs but only two distinct outputs.

Common mistake to avoid

Do not confuse a function with a one-to-one function. Also check for missing domain elements, not only repeated ones.

Try it yourself

1. Is {(1, 2), (1, 4), (2, 5)} a function? 2. Is {(1, 2)} a function on domain {1, 2}?

Answers and reasoning

1. No: input 1 has two outputs. 2. No: input 2 has no output. On domain {1} alone it would be a function.

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