Codomain vs. Range — What's the Difference?
By Fiza Rafique & Urooj Arif — Updated on May 8, 2024
The codomain is the set of possible output values for a function, whereas the range refers to the actual set of values the function produces.
Difference Between Codomain and Range
Table of Contents
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Key Differences
The codomain is a broader mathematical concept, representing all potential outputs a function can produce. While it's predefined in the function's definition, the range is dynamic and consists of all actual values the function outputs.
In mathematics, the codomain is explicitly stated as part of the function's definition. On the other hand, the range is determined after applying the function, giving insight into which codomain elements are actually utilized.
For instance, if a function is defined from integers to real numbers, the codomain is the set of real numbers. However, the range might be just the positive real numbers, depending on the function's rule.
The codomain often includes values that the function does not reach, whereas the range is limited to values that the function outputs. Thus, the range is always a subset of the codomain.
In practice, defining the codomain helps understand the function's structure and mathematical behavior. Identifying the range, however, clarifies the function's actual effect and limits.
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Comparison Chart
Definition
Set of potential outputs of a function
Set of actual values a function outputs
Role
Predefined in function definition
Computed after applying the function
Inclusivity
May include values not reached
Limited to values the function reaches
Mathematical Use
Provides theoretical structure
Offers practical insight into behavior
Subset Relation
Not necessarily a subset of the range
Always a subset of the codomain
Compare with Definitions
Codomain
The set of all potential outputs of a function.
For a function mapping integers to real numbers, the codomain is the set of all real numbers.
Range
Varies depending on the input values provided.
The range can change based on the input set.
Codomain
Part of a function's definition, detailing the possible result values.
The codomain is important for understanding function limitations.
Range
Useful for understanding the practical application of the function.
The range showed the limitations of the function's output.
Codomain
A crucial aspect of defining functions in mathematical notation.
The function was defined with a codomain including all integers.
Range
Constrained by the specific function rule applied to inputs.
By analyzing the rule, we can predict the function's range.
Codomain
All possible values to which the function can theoretically map inputs.
The codomain can be larger than the actual range.
Range
The actual set of output values a function generates.
The function's range is just the positive numbers.
Codomain
Theoretically determined by the nature of the function's mapping.
The codomain sets expectations for what a function can output.
Range
A subset of the codomain representing the function's true output.
Despite the large codomain, the range was much smaller.
Codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term range is sometimes ambiguously used to refer to either the codomain or image of a function.
Range
The area of variation between upper and lower limits on a particular scale
Grand hotels were outside my price range
The cost will be in the range of $1–5 million a day
Codomain
A set containing all the values of a function. The range is a subset of the codomain.
Range
A set of different things of the same general type
The company's new carpet range
The area offers a wide range of activities for the tourist
Codomain
The target set into which a function is formally defined to map elements of its domain; the set denoted Y in the notation f : X → Y.
Range
The distance within which a person can see or hear
Something lurked just beyond her range of vision
Range
A line or series of mountains or hills
A mountain range
Range
A large area of open land for grazing or hunting
On dude ranches, tourists put on crisp new western gear to ride the range
Range
A large cooking stove with burners or hotplates and one or more ovens, all of which are kept continually hot
A wood-burning kitchen range
Range
A row of buildings
Townesend's Durham quadrangle range at Trinity College
Range
The direction or position in which something lies
The range of the hills and valleys is nearly from north to south
Range
Vary or extend between specified limits
Prices range from £30 to £100
Range
Place or arrange in a row or rows or in a specified manner
A table with half a dozen chairs ranged around it
Range
Place oneself or be placed in opposition to (a person or group)
Japan ranged herself against the European nations
Range
(of a person or animal) travel or wander over a wide area
Patrols ranged deep into enemy territory
Free-ranging groups of baboons
Tribes who ranged the windswept lands of the steppe
Range
Obtain the range of a target by adjustment after firing past it or short of it, or by the use of radar or laser equipment
Radar-type transmissions which appeared to be ranging on our convoys
Range
(of a person or their lifestyle) orderly; settled
It's possible to be too rangé
By comparison with Strachey, Keynes was rangé
Range
A number or grouping of things in the same category or within specified limits
Offers a range of financial services.
Jobs at different pay ranges.
Range
An amount or extent of variation
A wide price range.
The range of genetic diversity.
Range
(Music) The gamut of tones that a voice or instrument is capable of producing. Also called compass.
Range
A class, rank, or order
The lower ranges of society.
Range
Extent of perception, knowledge, experience, or ability
Calculus is simply out of my range.
Range
The area or sphere in which an activity takes place
Beyond the range of the court's jurisdiction.
Range
The maximum extent or distance limiting operation, action, or effectiveness, as of a sound, radio signal, instrument, firearm, or aircraft
The limited range of the telescope.
Out of range of their guns.
Within hearing range.
Range
The maximum distance that can be covered by a vehicle with a specified payload before its fuel supply is exhausted.
Range
The distance between a projectile weapon and its target.
Range
A place equipped for practice in shooting at targets.
Range
A testing area at which rockets and missiles are launched and tracked.
Range
A place or business where golf shots can be practiced.
Range
An extensive area of open land on which livestock wander and graze.
Range
The geographic region in which a plant, animal, or other organism normally lives or grows.
Range
The opportunity or freedom to wander or explore
We had free range of the campus.
Range
(Mathematics) The set of all values a given function may take on.
Range
(Statistics) The difference or interval between the smallest and largest values in a frequency distribution or a set of data.
Range
A group or series of things extending in a line or row, especially a row or chain of mountains.
Range
One of a series of double-faced bookcases in a library stack room.
Range
A north-south strip of townships, each six miles square, numbered east and west from a specified meridian in a US public land survey.
Range
A stove with spaces for cooking a number of things at the same time.
Range
To vary within specified limits
Sizes that range from small to extra large.
Range
To extend in a particular direction
A river that ranges to the east.
Range
To cover or have application to a number of things
Their conversation ranged over the major issues of the day. Her responsibilities range across all aspects of the negotiations.
Range
To move through, along, or around in an area or region
Raiders ranged up and down the coast.
Range
To wander freely; roam
Allowed the animals to range freely.
Range
To look over something or around an area or place
The teacher's eyes ranged over the class.
Range
To live or grow within a particular region
"Some animals and plants range over a large portion of the world, yet retain the same character" (Charles Darwin).
Range
To arrange or dispose in a particular order, especially in rows or lines
"In the front seats of the galleries were ranged the ladies of the court" (Carolly Erickson).
Range
To assign to a particular category; classify
Her works are often ranged under the headings Mystery and Science Fiction.
Range
To move through or along or around in (an area or region)
The scouts ranged the mountain forests. The patrol boat ranged the coast.
Range
To look over or throughout (something)
His eyes ranged the room, looking for the letter.
Range
To turn (livestock) onto an extensive area of open land for grazing.
Range
To align (a gun, for example) with a target.
Range
To determine the distance of (a target).
Range
To be capable of reaching (a maximum distance).
Range
(Nautical) To uncoil (an anchor cable) on deck so the anchor may descend easily.
Range
A line or series of mountains, buildings, etc.
Range
A fireplace; a fire or other cooking apparatus; now specifically, a large cooking stove with many hotplates.
Range
Selection, array.
We sell a wide range of cars.
Range
An area for practicing shooting at targets.
Range
An area for military training or equipment testing.
Range
The distance from a person or sensor to an object, target, emanation, or event.
We could see the ship at a range of five miles.
One can use the speed of sound to estimate the range of a lightning flash.
Range
The maximum distance or reach of capability (of a weapon, radio, detector, etc.).
This missile's range is 500 kilometres.
Range
The distance a vehicle (e.g., a car, bicycle, lorry, or aircraft) can travel without refueling.
This aircraft's range is 15 000 kilometres.
Range
An area of open, often unfenced, grazing land.
Range
The extent or space taken in by anything excursive; compass or extent of excursion; reach; scope.
Range
(mathematics) The set of values (points) which a function can obtain.
Range
(statistics) The length of the smallest interval which contains all the data in a sample; the difference between the largest and smallest observations in the sample.
Range
The defensive area that a player can cover.
Jones has good range for a big man.
Range
(music) The scale of all the tones a voice or an instrument can produce.
Range
(ecology) The geographical area or zone where a species is normally naturally found.
Range
(programming) A sequential list of values specified by an iterator.
Calls the given function on each value in the input range.
Range
An aggregate of individuals in one rank or degree; an order; a class.
Range
(obsolete) The step of a ladder; a rung.
Range
A bolting sieve to sift meal.
Range
A wandering or roving; a going to and fro; an excursion; a ramble; an expedition.
Range
In the public land system, a row or line of townships lying between two succession meridian lines six miles apart.
Range
The variety of roles that an actor can play in a satisfactory way.
By playing in comedies as well as in dramas he has proved his range as an actor.
By playing in comedies as well as in dramas he has proved his acting range.
Range
(intransitive) To travel over (an area, etc); to roam, wander.
Range
(transitive) To rove over or through.
To range the fields
Range
To exercise the power of something over something else; to cause to submit to, over.
Range
(transitive) To bring (something) into a specified position or relationship (especially, of opposition) with something else.
Range
(intransitive) Of a variable, to be able to take any of the values in a specified range.
The variable x ranges over all real values from 0 to 10.
Range
(transitive) To classify.
To range plants and animals in genera and species
Range
(intransitive) To form a line or a row.
The front of a house ranges with the street.
Range
(intransitive) To be placed in order; to be ranked; to admit of arrangement or classification; to rank.
Range
(transitive) To set in a row, or in rows; to place in a regular line or lines, or in ranks; to dispose in the proper order.
Range
(transitive) To place among others in a line, row, or order, as in the ranks of an army; usually, reflexively and figuratively, to espouse a cause, to join a party, etc.
Range
(biology) To be native to, or live in, a certain district or region.
The peba ranges from Texas to Paraguay.
Range
To determine the range to a target.
Range
To sail or pass in a direction parallel to or near.
To range the coast
Range
(baseball) Of a player, to travel a significant distance for a defensive play.
Range
To set in a row, or in rows; to place in a regular line or lines, or in ranks; to dispose in the proper order; to rank; as, to range soldiers in line.
Maccabeus ranged his army by bands.
Range
To place (as a single individual) among others in a line, row, or order, as in the ranks of an army; - usually, reflexively and figuratively, (in the sense) to espouse a cause, to join a party, etc.
It would be absurd in me to range myself on the side of the Duke of Bedford and the corresponding society.
Range
To separate into parts; to sift.
Range
To dispose in a classified or in systematic order; to arrange regularly; as, to range plants and animals in genera and species.
Range
To rove over or through; as, to range the fields.
Teach him to range the ditch, and force the brake.
Range
To sail or pass in a direction parallel to or near; as, to range the coast.
Range
To be native to, or to live in; to frequent.
Range
To rove at large; to wander without restraint or direction; to roam.
Like a ranging spaniel that barks at every bird he sees.
Range
To have range; to change or differ within limits; to be capable of projecting, or to admit of being projected, especially as to horizontal distance; as, the temperature ranged through seventy degrees Fahrenheit; the gun ranges three miles; the shot ranged four miles.
Range
To be placed in order; to be ranked; to admit of arrangement or classification; to rank.
And range with humble livers in content.
Range
To have a certain direction; to correspond in direction; to be or keep in a corresponding line; to trend or run; - often followed by with; as, the front of a house ranges with the street; to range along the coast.
Which way the forests range.
Range
To be native to, or live in, a certain district or region; as, the peba ranges from Texas to Paraguay.
Range
A series of things in a line; a row; a rank; as, a range of buildings; a range of mountains.
Range
An aggregate of individuals in one rank or degree; an order; a class.
The next range of beings above him are the immaterial intelligences.
Range
The step of a ladder; a rung.
Range
A kitchen grate.
He was bid at his first coming to take off the range, and let down the cinders.
Range
An extended cooking apparatus of cast iron, set in brickwork, and affording conveniences for various ways of cooking; also, a kind of cooking stove.
Range
A bolting sieve to sift meal.
Range
A wandering or roving; a going to and fro; an excursion; a ramble; an expedition.
He may take a range all the world over.
Range
That which may be ranged over; place or room for excursion; especially, a region of country in which cattle or sheep may wander and pasture.
Range
Extent or space taken in by anything excursive; compass or extent of excursion; reach; scope; discursive power; as, the range of one's voice, or authority.
Far as creation's ample range extends.
The range and compass of Hammond's knowledge filled the whole circle of the arts.
A man has not enough range of thought.
Range
The region within which a plant or animal naturally lives.
Range
The horizontal distance to which a shot or other projectile is carried.
Range
In the public land system of the United States, a row or line of townships lying between two successive meridian lines six miles apart.
Range
See Range of cable, below.
Range
An area in which something acts or operates or has power or control:
The range of a supersonic jet
The ambit of municipal legislation
Within the compass of this article
Within the scope of an investigation
Outside the reach of the law
In the political orbit of a world power
Range
The limits within which something can be effective;
Range of motion
He was beyond the reach of their fire
Range
A large tract of grassy open land on which livestock can graze;
They used to drive the cattle across the open range every spring
He dreamed of a home on the range
Range
A series of hills or mountains;
The valley was between two ranges of hills
The plains lay just beyond the mountain range
Range
A place for shooting (firing or driving) projectiles of various kinds;
The army maintains a missile range in the desert
Any good golf club will have a range where you can practice
Range
The limits of the values a function can take;
The range of this function is the interval from 0 to 1
Range
A variety of different things or activities;
He answered a range of questions
He was impressed by the range and diversity of the collection
Range
The limit of capability;
Within the compass of education
Range
A kitchen appliance used for cooking food;
Dinner was already on the stove
Range
Change or be different within limits;
Estimates for the losses in the earthquake range as high as $2 billion
Interest rates run from 5 to 10 percent
The instruments ranged from tuba to cymbals
My students range from very bright to dull
Range
Move about aimlessly or without any destination, often in search of food or employment;
The gypsies roamed the woods
Roving vagabonds
The wandering Jew
The cattle roam across the prairie
The laborers drift from one town to the next
They rolled from town to town
Range
Have a range; be capable of projecting over a certain distance, as of a gun;
This gun ranges over two miles
Range
Range or extend over; occupy a certain area;
The plants straddle the entire state
Range
Lay out in a line
Range
Feed as in a meadow or pasture;
The herd was grazing
Range
Let eat;
Range the animals in the prairie
Range
Assign a rank or rating to;
How would you rank these students?
The restaurant is rated highly in the food guide
Common Curiosities
Is the codomain always larger than the range?
The codomain may include values that the function doesn't reach, so it is typically larger than the range.
Why do functions need a codomain?
The codomain sets expectations for potential outputs, helping define and analyze the function's behavior.
Does the range depend on the function's input?
Yes, the range is directly affected by the inputs used in the function, as it consists of the actual outputs from these inputs.
Can changing the codomain affect the function?
Changing the codomain doesn't affect the function's operation but can change how it's perceived, particularly regarding surjectivity.
How can I identify the range of a function?
To find the range, evaluate the function with different inputs to see the actual outputs.
How do you determine the codomain of a function?
The codomain is determined when the function is defined, based on the possible outputs that fit the nature of the function.
What happens if the range equals the codomain?
If the range equals the codomain, the function is considered surjective, or onto, meaning every element of the codomain is mapped by some input.
Why is it important to distinguish between codomain and range?
Distinguishing these helps in understanding the types of functions (like injective, surjective) and their behaviors in different mathematical contexts.
Can the codomain and range be the same set?
Yes, when a function reaches every value in the codomain, the range matches the codomain.
Is the range always a subset of the codomain?
Yes, by definition, the range must always be a subset of the codomain.
Can the range be empty?
The range cannot be empty as long as the function has an input set; there must be at least one output.
Is it possible for the range to become the entire codomain?
Yes, this is possible if every possible output in the codomain is reached, making the function surjective.
How do we use codomain and range in real-world applications?
In programming, functions must have expected output types (codomain) and the actual outputs (range) which are crucial for debugging and functionality checks.
What tools are used to identify the range of complex functions?
Mathematical analysis and graphing are common tools used to determine the range of more complex functions.
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Written by
Fiza RafiqueFiza Rafique is a skilled content writer at AskDifference.com, where she meticulously refines and enhances written pieces. Drawing from her vast editorial expertise, Fiza ensures clarity, accuracy, and precision in every article. Passionate about language, she continually seeks to elevate the quality of content for readers worldwide.
Co-written by
Urooj ArifUrooj is a skilled content writer at Ask Difference, known for her exceptional ability to simplify complex topics into engaging and informative content. With a passion for research and a flair for clear, concise writing, she consistently delivers articles that resonate with our diverse audience.
