A function is a rule that assigns each input exactly one output. We call the output the image of the input. The set of all inputs for a function is called the domain. The set of all allowable outputs is called the codomain. We would write \(f:X \to Y\) to describe a function with name \(f\text{,}\) domain \(X\) and codomain \(Y\text{.}\) This does not tell us which function \(f\) is though. To define the function, we must describe the rule. This is often done by giving a formula to compute the output for any input (although this is certainly not the only way to describe the rule).
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