DOMAIN AND CONTRADOMAIN OF A FUNCTION (WITH EXAMPLES) - MATHS - 2025

The concepts of domain and counter-domain of a function are commonly taught in calculus courses that are taught at the beginning of university degrees.

Before defining the domain and the contradomain, you must know what a function is. A function f is a law (rule) of correspondence made between the elements of two sets.

The set from which the elements are chosen is called the domain of the function, and the set to which these elements are sent through f is called the counter-domain.

In mathematics a function with domain A and counter domain B is denoted by the expression f: A → B.

The previous expression says that the elements of set A are sent to set B following the correspondence law f.

A function assigns each element of set A a single element of set B.

Domain and contradomain

Given a real function of a real variable f (x), we have that the domain of the function will be all those real numbers such that, when evaluated in f, the result is a real number.

Generally, the counter-domain of a function is the set of real numbers R. The counter-domain is also called the arrival set or codomain of the function f.

Is the contradomain of a function always R?

No. As long as the function is not studied in detail, the set of real numbers R is usually taken as a counter-domain.

But once the function has been studied, a more suitable set can be taken as a counter-domain, which will be a subset of R.

The proper set that was mentioned in the previous paragraph matches the image of the function.

The definition of the image or range of a function f refers to all the values ​​that come from evaluating an element of the domain in f.

Examples

The following examples illustrate how to calculate the domain of a function and its image.

Example 1

Let f be a real function defined by f (x) = 2.

The domain of f is all real numbers such that, when evaluated at f, the result is a real number. The contradomain for the moment is equal to R.

As the given function is constant (always equal to 2), it does not matter which real number is chosen, since when evaluating it in f the result will always be equal to 2, which is a real number.

Therefore, the domain of the given function is all real numbers; that is, A = R.

Now that it is known that the result of the function is always equal to 2, we have that the image of the function is only the number 2, therefore the counter-domain of the function can be redefined as B = Img (f) = {two}.

Therefore, f: R → {2}.

Example 2

Let g be a real function defined by g (x) = √x.

As long as the image of g is not known, the contradomain of g is B = R.

With this function it must be taken into account that the square roots are only defined for non-negative numbers; that is, for numbers greater than or equal to zero. For example, √-1 is not a real number.

Therefore, the domain of the function g must be all numbers greater than or equal to zero; that is, x ≥ 0.

Therefore, A = [0, + ∞).

To calculate the range, it should be noted that any result of g (x), because it is a square root, will always be greater than or equal to zero. That is, B = [0, + ∞).

In conclusion, g: [0, + ∞) → [0, + ∞).

Example 3

If we have the function h (x) = 1 / (x-1), we have that this function is not defined for x = 1, since the denominator would obtain zero and the division by zero is not defined.

On the other hand, for any other real value the result will be a real number. Therefore, the domain is all reals except one; that is, A = R {1}.

In the same way, it can be observed that the only value that cannot be obtained as a result is 0, since for a fraction to be equal to zero the numerator must be zero.

Therefore, the image of the function is the set of all reals except zero, so B = R {0} is taken as a contradomain.

In conclusion, h: R {1} → R {0}.

Observations

The domain and the image do not have to be the same set, as demonstrated in Examples 1 and 3.

When a function is graphed on the Cartesian plane, the domain is represented by the X axis and the counterdomain or range is represented by the Y axis.

References

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