A relation is a set of ordered pairs or a correspondence between inputs and outputs; it is a function when every input in its domain is associated with exactly one output. The distinction is interpreted in mappings, tables, graphs, and equations, including use of the vertical-line test, while recognizing that different inputs may share the same output and that a function need not be one-to-one. This scope concerns single-input relations represented in standard algebraic forms, not more abstract or multivariable function theory.
A relation pairs inputs with outputs, usually written as ordered pairs . It is a function only when each input has exactly one output.
Consider the relation
Step 1: Identify the inputs.
The inputs are the first coordinates:
Step 2: Check repeated inputs.
The input is paired with both and :
The input is also paired with two different outputs:
Step 3: Decide whether it is a function.
Because some inputs have more than one output, is not a function. It is still a relation.
On a graph, this same conclusion comes from the vertical-line test: a vertical line at or would cross the graph twice. Any graph that a vertical line crosses more than once does not represent a function.
Remember that different inputs may share the same output. For example, and would not violate the function rule. The rule only prohibits one input from having multiple outputs.
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