Explain why your relation is a function
The Correct Answer and Explanation is :
In mathematics, a relation is defined as a function if each input (or element from the domain) is associated with exactly one output (or element from the range). This means that for every value of the independent variable (input), there is only one corresponding value of the dependent variable (output).
To clarify why a relation might be a function, let’s explore an example:
Example: Suppose we have the relation ({(1,2), (2,4), (3,6)}). This is a function because each input (1, 2, 3) is paired with exactly one output (2, 4, 6). No input repeats with different outputs.
Now, let’s look at why this relation satisfies the definition of a function:
- Unique outputs for each input: The key criterion for a relation to be a function is that each input must map to a unique output. In our example, the input 1 maps to 2, 2 maps to 4, and 3 maps to 6. No input is assigned to more than one output.
- Vertical Line Test: This is a graphical method used to determine if a relation is a function. If a vertical line can be drawn through the graph of the relation and it intersects the graph in more than one point, the relation is not a function. In the case of our example, if plotted, each input would have only one output, and no vertical line would intersect the graph more than once.
Therefore, the relation ({(1,2), (2,4), (3,6)}) is a function because each input is uniquely related to a single output, fulfilling the definition of a function.