Definition
Function
A function is a relationship that assigns exactly one output to each allowed input.
What it means
A function never changes its mind about the same input. Put in an allowed value and the rule gives one definite output. Two different inputs are allowed to land on the same output, but one input cannot split into two outputs in the same function.
Remember it
Same input, same single output.
Example
For f(x) = x², input 3 gives output 9. Input −3 also gives 9, which is allowed. But input 3 cannot also give 8 in that same function.
Watch for
A rule that sends input 2 to both 5 and 7 is not a function, because one input has two outputs.
Common mix-up
A function does not require every output to be different; only each input must have exactly one output.
Exact meaning
A function is a relation that assigns each allowed input exactly one output. Different inputs may share an output, and the function may be represented by a rule, table, graph, mapping, or formula.