Functions: Definitions and Types

Functions: Definitions and Types

Functions on sets:

A function F (or f) between A and B is a relation which satisfies the following conditions.

  1. The domain of F must be equal to A i.e., every element of A must occur at least in one pair
  2. Every element of the domain must be related to one and only one element of the range.  i.e., if aRc and aRd then c=d.  That is, a is related to only one element of B or equivalently, every element of A must occur in only one pair and not more than one pair.

Equivalently,

A function is a subset of AXB in which every element of A must occur in one and only one ordered pair.  Some elements of B may not occur and one element of B may occur more than once in the ordered pairs.

If A = {a, b} and B = {1, 2 } then f  = {(a, 2), (b, 2)} is a function on A since

1.All elements of A occurs in at least one pair.

2.All elements of A occurs in only one pair.

3.There is no need that all elements of B must occur.

f1 = {(a, 2), (b, 2), (a, 1)} is not a function on A since
  1. All elements of A occur in at least one pair.
  2. But a ∈ A occurs in more than one pair
f1 is not a function since it does not satisfy condition2 for functions.

Representation of Functions:

If A={a, b} , B={1, 2} take the function f  = {(a, 2), (b, 2)} on A.  Symbolically, function f is written as,

𝑓 :𝐴→𝐵

and read as f from A to B; Where A is the domain of f and range of f is a subset of B.  The set representing f is replaced by the assignments

𝑓(a)=2 and 𝑓(b)=2

and read as f of a is equal to 2 and f of b is equal to 2. Elements of the domain  A are said to be the arguments of f and the related element of the range is said to be its value or image. Functions may also be called mappings.

For example, in f(a)=2 , a is an argument of f and 2 is said to be the image (or value) of a in B.

Inverse of a function:

The function which has the range of 𝑓 as its domain and the domain of 𝑓 as its range is said to be the inverse function of 𝑓 and is written as 𝑓-1.  In other words, if 𝑓 is a function from A into B then 𝑓-1 is a function from range(𝑓)⊆B onto A.

i.e., if 𝑓 :𝐴→𝐵 then 𝑓-1:range(𝑓)→A

For A={a, b} , B={1, 2} consider

𝑓= {(a,2), (b,2)}⊆AXB, i.e., 𝑓(a)=2 and 𝑓(b)=2

so the range of 𝑓  is the singleton set {2}⊆B because 1 is not related to any element of A.  Clearly, 𝑓 is a function since both a and b elements of A exists in one and only one pair.  Then 𝑓-1 is given by,

𝑓-1:{2}→A with 𝑓-1(2) = a and 𝑓-1(2) = b

a and b are said to be the inverse images of 2 and written as,

𝑓-1(2) = {a, b} = A

sample
Here 𝑓-1 is a relation but not a function. The inverse of 𝑓 will also be a function iff 𝑓 is a 1-1 correspondence between A and B.

Image and Inverse image of sets:

Image of a set A is the set of all images of elements of A.

𝑓(A) = {𝑓(x)| ∀ x∈A}

The inverse image of a set B is the set of all inverse images of elements of B. Symbolically,

𝑓 -1(B) = {𝑓 -1(y)| ∀ y∈B}

If A={a1, a2,a3} and B={b1,b2,b3} then

for 𝑓 :𝐴→𝐵 defined by 𝑓(ai) = bi then 𝑓(A)={𝑓 (a1), 𝑓 (a2), 𝑓 (a3)} = {b1,b2,b3} = B and 𝑓 -1(B)={𝑓 -1 (b1), 𝑓 -1 (b2), 𝑓 -1 (b3)} = {a1,a2,a3} = A

Here both 𝑓 and 𝑓 -1 are 1-1 correspondence between A and B and so |A| = |B|.

Types of Functions:

A function 𝑓 from A to A

𝑓 :𝐴→𝐴

is said to be an identity function if 𝑓 maps the element to itself.

i.e., 𝑓(x) = x for every x∈A

Some times IA may also be used to denote identity function on A.

A function 𝑓 from A to B

𝑓 :𝐴→𝐵

is said to be an into function if the range of 𝑓 is a subset of B.  For example, if A={a, b} and B={1, 2} then the function 𝑓 defined by 𝑓 = {(a, 2), (b, 2)} on A is an into function since

range of 𝑓 = {2} ⊆ B

This function can be read as 𝑓 from A into B.  In into functions at least one element of B will not occur in any pair represent 𝑓.
A function 𝑓 from A to B

𝑓 :𝐴→𝐵

is said to be an onto function if the range of 𝑓 is the set B.  For example, if A={a, b} and B={1, 2} then the function 𝑓 defined by 𝑓 = {(a, 2), (b, 1)} on A is an onto function since

range of 𝑓 = {2, 1} = B

This function can be read as 𝑓 from A onto B.  In onto functions every element of B occurs in at least one pair represent 𝑓.

A function 𝑓 from A to B

𝑓 :𝐴→𝐵

is said to be an 1-1 (one to one) function if every element of the range of 𝑓 will be an image of only one element of A.  For example, if A={a, b} and B={1, 2} then the function 𝑓 defined by 𝑓 = {(a, 2), (b, 1)} on A is an 1-1 function since

  • 1∈B is an image of only one element and
  • 2∈B is an image of only one element

A function 𝑓 from A to B

𝑓 :𝐴→𝐵

is said to be a bijection if it is both 1-1 and onto.  Then 𝑓 is said to be a bijection from A onto B or an 1-1 correspondence between A and B.  Then, |A| = |B|.

Composition of Functions:

If 𝑓 is a function from A to B and g is a function from B to C then the Composition of 𝑓 and is a function from A to C which maps an element ‘a’ of A to an element of C which is nothing but the image of ‘𝑓(a)’ under g.  The composition of 𝑓 and g is denoted by g ∘ 𝑓.

If A={a1,a2,a3}, B={b1,b2,b3} and C={c1,c2,c3} then

for 𝑓 :𝐴→𝐵 defined by 𝑓(ai) = bi and

for g :B→C defined by g(bi) = ci then

(g ∘ 𝑓)(ai) = g(𝑓(ai)) = g(bi) = ci

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If g = 𝑓 -1 then (g ∘ 𝑓)(ai) = (𝑓 -1 ∘ 𝑓)(ai) = 𝑓 -1(𝑓(ai)) = 𝑓 -1(bi) = ai = IA(ai). Therefore for any function 𝑓, (𝑓 -1 ∘ 𝑓) = I, the identity function on the domain of 𝑓.

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