A transfer function is an intermediate function that comprises & explains the theoretical procedures on how the input is transferred to the relevant output.
It is applicable from simple algebraic sums, engineering applications to everyday phenomena.
Examples
In Y=2a−b×x, the input x is transfered to its output Y according to the transfer function 2a−b
How water at room temperature boils upon the received heat is the transfer function of that phenomenon.
But a transfer function is usually described only for engineering applications such as control systems & yet it is involved in every cause & result system.
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