4.1.3 • Examining Relations and Functions
Activity
A relation is any set of ordered pairs, .
A function is a special type of relation that maps every input, , to exactly one output, .
Tell if each relation is a function and explain.
1.
| -3 | 3 |
| -2 |
2 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
Compare your answer:
Yes, this is a function because every input goes to exactly one output.
2.
Compare your answer:
This relation is not a function. Raul and Jenny have 2 emails, so twice there are inputs that go to two outputs.
3.
Compare your answer:
This is not a function because each input goes to multiple outputs.
4.
Compare your answer:
Each input, -value, is matched with only one output, -value. So this relation is a function.
5.
Compare your answer:
The input (-value), 9, is matched with two outputs (-values), both 3 and –3. So this relation is not a function.
Self Check
Additional Resources
Is the Relation a Function?
A relation is a set of ordered pairs. The set of the first components of each ordered pair are the inputs and the set of the second components of each ordered pair are the outputs. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.
The set of inputs is {}. The set of outputs is {}.
Each input value is the independent variable, and is often labeled with the lowercase letter . Each output value, is the dependent variable, and is often labeled lowercase letter .
A function is a relation that assigns a single output to each input. In other words, no -values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each input, {1,2,3,4,5}, is paired with exactly one output, {}.
Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as
{ (odd,1),(even,2),(odd,3),(even,4),(odd,5) }
Notice that each input, {even,odd} is not paired with exactly one output, {}. For example, the term “odd” corresponds to three output values, {1,3,5} and the term “even” corresponds to two output values, {}. This violates the definition of a function, so this relation is not a function.
A mapping is sometimes used to show a relation. The arrows show the pairing of the inputs with the outputs.
Here are some relations that are functions and not functions.
- This relationship is a function because each input is associated with a single output. Note that inputs and both give output .
- This relationship is also a function. In this case, each input is associated with a single output.
- This relationship is not a function because input is associated with two different outputs.
How to determine if a relationship is a function.
- Identify the input values.
- Identify the output values.
- If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.
Try it
Is the Relation a Function?
Is the following relation a function? Explain your answer.
Here is how to determine if this relation is a function:
Since every input (name) goes to only one output (birthday), this relation is a function.