Function

Relation is a very broad concept. It has various forms in a very natural way. To make it more useful and concrete, it is necessary to refine it. Let’s begin with a relation ℜ on a set A and an ordered pair (a, b) ∈ ℜ. A refinement of ℜ can be obtained by assigning only one second element ‘b’ to each first element ‘a’ of the pair (a, b). Such a refinement of a relation is known as function.

Consider a function defined from set A to another set B or the same set A. If x∈A, i.e.,

A={x:x satisfies a certain property}, the letter or symbol, x is known as a variable. Each number or element of the set A is called the value of the variable x. If x∈A={1, 2, 3}, then x is a variable which stands for one of the members 1, 2, 3. Here, each of the numbers 1, 2, 3 is a value of the variable x.

Consider a singleton set C={c} i.e. a set C consisting of only one member c, then the letter or symbol c is known as a constant. In other words, a constant has a fixed value.

Consider two sets A and B. Then, any non-empty subset ℜ of the Cartesian product A×B is known as a relation from A to B. Refinement of this relation can be done by associating each element of the set A with a unique element of B.

Example 1:

Let A={1, 2, 3} and B={2, 4, 6}.

A×B={(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 2), (3, 4), (3, 6)}

Let us make the following selection (or refinement): {(1, 2), (2, 4), (3, 6)}. This is a refined relation from A to B. Here, each first element is uniquely associated with a second element. The rule or relation corresponding to the present assignment is, “To each first element, there corresponds a unique second element that is twice the first element”.

In tabular form:

Example 1: Tabular form, Function

Graphically,

Example 1: Graphically, Function

In arrow diagram:

Example 1: Arrow Diagram, Function

Example 2:

Let A={a, b, c, d} and B={x, y, z, t}.

A×B={(a, x), (a, y), (a, z), (a, t), (b, x), (b, y), (b, z), (b, t), (c, x), (c, y), (c, z), (c, t), (d, x), (d, y), (d, z), (d, t)}

Let us make the following selection: {(a, x), (b, y), (c, z), (d, z)}. This is a refined relation from A to B. Here, each first element is uniquely associated with a second element. But there is a slight difference, there are two first elements that correspond to one second element.

In tabular form:

Example 2: Tabular form, Function

In arrow diagaram:

Example 2: Arrow diagram, Function

Although every first element is associated with a unique second element, there is one element of the second set b that is not associated with any element of A.

A relation is said to be a function from set A to set B if every element of set A associates with a unique element of set B.

In other words, a function from a set A to a set B is a relation ƒ such that for each a∈A, ∃ a unique b∈B s.t. (a, b)∈ƒ.

Here, “∃” is read “there exists” and “s.t.” stands for “such that”.

Also, a function from a set A to a set B is a relation in which no two ordered pairs have the same first co-ordinate.

A function ƒ from a set A to a set B is denoted by, \[ƒ:A \to B\] \[\text{or, } A \xrightarrow{ƒ} B\] if y∈B is associated with an element x∈A, we write it as, \[y=ƒ(x)\] which is read “y equals f of x”. ƒ(x) is known as the image of ƒ at x or value of ƒ at x.

If b is the unique element of B corresponding to an element a of A under the function ƒ, then a is called the pre-image of b under ƒ and b is called the image of a under ƒ. \[b=ƒ(a)\] The element y∈B with which the element x∈A associates, is known as the image of x under ƒ. The element x∈A which associates with y∈B, is known as the pre-image of y under ƒ.

It is to be noted that if ƒ is a function from A to B, then no element of A is related to more than one element of B; although more than one element of A can be related to the same element of B. In each case, A is called the domain of ƒ and B its co-domain.

The subset of B that contains only those elements of B that have pre-images in A is called the range of ƒ. \[\text{Range of }ƒ⊆B\] Therefore, in a function ƒ: A➞B, the set of values of x∈A for which the function is defined, is said to be the domain of the function. It is denoted by dom(ƒ) or simply D(ƒ).

∴ D(ƒ) = {x:x∈A for which ƒ(x) is defined}

The set of values of y=ƒ(x)∈B for every x∈A is known as the range of the function. It is denoted by range(ƒ) or simply R(ƒ).

∴ R(ƒ) = {y:y∈B, y=ƒ(x) for all x∈A}

Example 3: \[y=ƒ(x)=\sqrt{x-1}\] then, ƒ(x) is defined for x-1≥0 i.e. x≥1. Thus, \[D(ƒ)=\{x:x≥1\}=[1,\infty)\] \[R(ƒ)=\{y:y≥0\}=[0,\infty)\] In the function ƒ:A➞B, A itself is the domain of ƒ and the set of values of y∈B which are the images of the elements of A, is the range of ƒ.

A function is also called a mapping. In some special context, a transformation or an operator.

Since y is the image of an element x under a function ƒ, ƒ is also indicated by \[x \mapsto y \text{ or, } x \mapsto ƒ(x)\] In the above example: 2 where A={a, b, c, d} and B={x, y, z, t} and the refinement, \[\{(a,x),(b,y),(c,z),(d,z)\}\] the function ƒ:A➞B may be described by equations \[ƒ(a)=x,ƒ(b)=y,ƒ(c)=z \text{ and } ƒ(d)=z\] \[\therefore D(ƒ)=\{a,b,c,d\}\] \[R(ƒ)=\{x,y,z\}\] Here, \[ƒ(A)≠B \text{ and } ƒ(c)=ƒ(d)=z\] If two functions ƒ and g: A➞B have a same domain D, then they are said to be equal iff ƒ(x)=g(x) for every x∈A⊆D; and is written as ƒ=g.

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