Formal Definition
Standard mathematical notation allows a formal definition of range.
In the first sense, the range of a function must be specified; it is often assumed to be the set of all real numbers, and {y | there exists an x in the domain of f such that y = f(x)} is called the image of f.
In the second sense, the range of a function f is {y | there exists an x in the domain of f such that y = f(x)}. In this case, the codomain of f must be specified, but is often assumed to be the set of all real numbers.
In both cases, image f ⊆ range f ⊆ codomain f, with at least one of the containments being equality.
Read more about this topic: Range (mathematics)
Famous quotes containing the words formal and/or definition:
“That anger can be expressed through words and non-destructive activities; that promises are intended to be kept; that cleanliness and good eating habits are aspects of self-esteem; that compassion is an attribute to be prized—all these lessons are ones children can learn far more readily through the living example of their parents than they ever can through formal instruction.”
—Fred Rogers (20th century)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)