Lesson · Grade 10 · Foundations of Mathematics and Pre Calculus
Relations and Functions: One Output for Each Input
- Grade 10
- Foundations of Mathematics and Pre Calculus
- 1 min 55 s
- British Columbia
- BC10-MA-07
Relation A = {(1, 4), (2, 5), (3, 4)} is written as ordered pairs, drawn as a mapping diagram and checked input by input. Each input has one arrow, so it is a function even though inputs 1 and 3 share the output 4, just as two people can share a birthday. Relation B = {(1, 4), (1, 6), (2, 5)} is drawn beside it: input 1 has two outputs, which you can see both in the diagram and in the pairs, so it is not a function, just as one birthday can belong to two people.
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In one sentence
The question this lesson answers
When is a relation also a function? Let's find out.
A relation is a set of ordered pairs, like relation A: 1 and 4, 2 and 5, and 3 and 4.
Transcript
What the lesson says
When is a relation also a function? Let's find out.
A relation is a set of ordered pairs, like relation A: 1 and 4, 2 and 5, and 3 and 4.
In each ordered pair, the first number is the input, and the second number is the output.
A mapping diagram puts the inputs in one oval and the outputs in another.
Each pair becomes an arrow from input to output: 1 to 4, 2 to 5, and 3 to 4.
Count the arrows from each input: 1 has one arrow, 2 has one, and 3 has one.
A function is a relation in which each input has exactly one output. So relation A is a function.
Inputs 1 and 3 both go to 4, and that is allowed: two inputs may share an output.
People and birthdays work the same way: each person has one birthday, even if two people share it.
Now test relation B, with the pairs 1 and 4, 1 and 6, and 2 and 5.
In its mapping diagram, the arrows go from 1 to 4, from 1 to 6, and from 2 to 5.
Input 1 has two arrows, to 4 and to 6, so it has two outputs. Relation B is not a function.
In the ordered pairs, the input 1 appears twice, with two different outputs.
Birthdays to people may not be a function, because one birthday can belong to two people.
So a function gives exactly one output for each input. Outputs may repeat, but no input can have two.
Curriculum
Where this fits
- British Columbia, Grade 10
Foundations of Mathematics and Pre Calculus 10, Functions: functions and relations: connecting data, graphs, and situationsBC10-MA-07
The outcomes and strands follow the British Columbia curriculum.
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