# DOMAIN CODOMAIN AND RANGE OF A FUNCTION

Domain Codomain and Range of a Function :

Let f be a function and we have

f : A ---> B

Then, A is called the domain, B is called the co-domain of f.

The set f(A)  =  {f(x) : x ∈ A} is called the range.

## Domain Co-domain and Range of a Function - Example

Let A  =  {1, 2, 3, 4} and B  =  {1, 4, 9, 16, 25}

Let us consider the rule f(x)  =  x2 to map elements from A to B.

Then, we have

f(1)  =  12  =  1

f(2)  =  22  =  4

f(3)  =  32  =  9

f(4)  =  42  =  16

Clearly each element in A has a unique image in B.

So, f : A ---> B : f(x)  =  x2 is a function from A to B.

Here, we have

Domain (f)  =  A  =  {1, 2, 3, 4}

Co-Domain (f)  =  B  =  {1, 4, 9, 16, 25}

Range (f)  =  {1, 4, 9, 16}

Note :

If co-domain and range are equal, then the function will be an onto or surjective function.

If range is a proper subset of co-domain, then the function will be an into function.

## Domain Co-domain and Range of a Function - Practice Problems

Problem 1 :

Let A  =  {1, 2, 3} and B  =  {5, 6, 7, 8}.

f is the function which maps the elements from A to B as shown below.

Find, the domain, co-domain and range of f.

Solution :

Domain (f)  =  A  =  {1, 2, 3}

Co-domain (f)  =  B  =  {5, 6, 7, 8}

Range (f)  =  {5, 6, 7}

Problem 2 :

Let X  =  {a, b, c, d} and Y  =  {d, e, f}.

f is the function which maps the elements from X to Y as shown below.

Find, the domain, co-domain and range of f.

Solution :

Domain (f)  =  A  =  {a, b, c, d}

Co-domain (f)  =  B  =  {d, e, f}

Range (f)  =  {d, e, f}

Problem 3 :

Let A  =  {1, 2, 3} and B  =  {5, 6, 7, 8}.

f is the function which maps the elements from A to B as shown below.

Find, the domain, co-domain and range of f.

Solution :

Domain (f)  =  A  =  {1, 2, 3, 4}

Co-domain (f)  =  B  =  {5, 6, 7, 8}

Range (f)  =  {5, 6, 7, 8}

Problem 4 :

Let X  =  {a, b, c} and Y  =  {d, e, f, g}.

f is the function which maps the elements from X to Y as shown below.

Find, the domain, co-domain and range of f.

Solution :

Domain (f)  =  A  =  {a, b, c}

Co-domain (f)  =  B  =  {d, e, f, g}

Range (f)  =  {e}

After having gone through the stuff given above, we hope that the students would have understood domain, co-domain and range of a function.

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