abrange

Abrange is a term used in some mathematical texts to denote the range bounds of a function over a given domain. It is not a standard term in mainstream mathematics, and its meaning can vary by author.

In one common usage, abrange(f, D) is defined as the closed interval [m, M], where m = inf{ f(x) : x in D } and M = sup{ f(x) : x in D }. This yields the smallest interval that contains all values of f on D. If f(D) is unbounded below or above, abrange is taken to be (-∞, ∞).

Example: let f(x) = x^2 and D = [-1, 2]. Then f(D) = [0, 4], so abrange(f, D) = [0, 4]. Note that abrange may be larger than the image if f(D) is not itself an interval.

Relation to range and image: abrange is sometimes used as a coarse bound rather than the exact range; it is distinct from the mathematical range or image, which is the actual set of outputs.

Abrange is a supplemental concept rather than a replacement for the standard notions of range or image. It is often used to discuss bounds on outputs in teaching, analysis, or interval-based reasoning, and it can be helpful when the precise image is difficult to determine. Related concepts include range, image, and interval arithmetic.