In mathematics, a **partial function** from *X* to *Y* is a function *ƒ: X' → Y*, where *X'* is a subset of *X*. It generalizes the concept of a function by not forcing *f* to map *every* element of *X* to an element of *Y* (only some subset *X*' of *X*). If *X'* = *X*, then *ƒ* is called a **total function** and is equivalent to a function. Partial functions are often used when the exact domain, *X'*, is not known (e.g. many functions in computability theory).

Specifically, we will say that for any *x* ∈ *X*, either:

*ƒ*(*x*) =*y*∈*Y*(it is defined as a single element in*Y*) or*ƒ*(*x*) is undefined.

For example we can consider the square root function restricted to the integers

Thus *g*(*n*) is only defined for *n* that are perfect squares (i.e. 0, 1, 4, 9, 16, ...). So, *g*(25) = 5, but *g*(26) is undefined.

Read more about Partial Function: Domain of A Partial Function, Total Function, Discussion and Examples

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### Famous quotes containing the words function and/or partial:

“The information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the *function* of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.”

—Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)

“You must not be *partial* in judging: hear out the small and the great alike; you shall not be intimidated by anyone, for the judgment is God’s.”

—Bible: Hebrew, Deuteronomy 1:17.