Functions

Relation
A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B.
Example:

Let A = {1,2,3,4} and B = {2,3,4,5,6} and a relation R: A → B defined by R = {(x,y):y=x+1}.

R = {(1, 2), (2, 3), (3, 4), (4, 5)}

Domain of relation R = A

Codomain of relation R = B

Range of relation R = {2, 3, 4, 5}

Range ⊂ Co-domain

Statements
Let X be a set of books in a shop. Y is the set of prices that correspond to each of the books in the shop. We have a correspondence between the books and their prices.

Let N be the set of natural numbers and C = {1, 8, 27, 64 …………..}. We can assign each natural number to its cube, giving a correspondence between the two sets, N and C.

In both the examples, we have two sets and a rule that assigns to every member of the first set a unique element of the second set.

Function is a type of a relation that assigns to every member of one set a unique element of another set.

Every function is a relation, but every relation is not a function.

Example 1: Let A = {1,2,3,4} and B = {2,3,4,5,6}, and a relation R: A → B defined by R = {(x,y):y=x+1}.
R = {(1, 2), (2, 3), (3, 4), (4, 5)}
Observe that all the elements of set A have a correspondence with the elements of set B.
Every element of set A has only one correspondence with the elements of set B.
Such type of relations in which every element in the domain set has one and only one image in set B are called functions. Hence, R is a function.

Let A and B be two non-empty sets. A function f : AàB, is a rule that maps each member of set A with a unique member of set B.

A relation R from a non-empty set A to another non-empty set B is said to be a function if its domain is set A and no two ordered pairs of R have the same first coordinates.

Example 2: f = {(4,6), (3,9), (-11,6), (3,11)}

(3,9) Î f and (3,11) Î f

Observe that both the ordered pairs have the same first coordinates.

Hence, the given relation is not a function.

Example 3:

f = {(x, x): x is a real number}

In this relation, for all real numbers, there exists an image that is the same real number.

Hence, the given relation is a function.

f is relation from set A to set B and f is said to be a function from A to B if, for all x ÎA, there exists a unique y ÎB such that (x, y)Î f .

Let A = {1, 2, 3, 4} and B = {2, 3, 4, 5, 6}, and a function f: A → B defined by f = {(x, y): y = x+1}

or, f = {(1, 2), (2, 3), (3, 4), (4, 5)}

Domain = A and Codomain = B

If f: AàB is a function and (x, y) Î f, then y is called the image of x under the function f and we denote this as f(x) = y.

Also, x is known as the pre-image or inverse image of y.

(1, 2), (2, 3), (3, 4), (4, 5) ∈ f

f(1) = 2, 2 is called the image of 1, or 1 is called the pre-image or inverse image of 2.

f(2) = 3, 3 is called the image of 2, or 2 is called the pre-image or inverse image of 3.

f(3) = 4, 4 is called the image of 3, or 3 is called the pre-image or inverse image of 4.

and f(4) = 5, 5 is called the image of 4, or 4 is called the pre-image or inverse image of 5.


Range

If f: AàB is a function, then the set of all the images of elements of set A is known as the range of the function f. It is denoted by f (A).

The range of a function is a subset to the codomain of the function.

f(A) = { 2, 3, 4, 5} and f(A) ⊆ B

In y = f(x), x is called the independent variable, while y is known as the dependent variable. The dependent variable is also called the value of function f at x.

Example:

Let f: AàR be a function defined by f(x) = 2x + 1, where A = {1, 2, -1}.

f(1) = 2(1) + 1 = 3

f(2) = 2(2) + 1 = 5

f(-1) = 2(-1) + 1 = -1

f = { (1, 3), (2, 5), (-1,-1)}


Vertical Line Test

The vertical line test helps us to check whether a given relation is a function or not.

If the vertical line intersects the graph of a elation at more than one place, it means that two ordered pairs have same first coordinates. Then we can conclude that it is not a function.

Algebra of functions

A function that has either R (the set of real numbers) or one of its subsets as its range is called a Real Valued Function.

If the domain and the range of the function is either R or a subset of R, then it is called a Real Function.

Addition of two functions:

If f: X à R and g : X à R are two real functions, where X ⊂ R, then f + g: X à R is defined as (f + g)(a) = f(a) + g(a), "a Î X.

Example: If f = {(1, 2), (2, 3), (3, 4)} and g = {(1, 5), (2, 6), (3, 7)}, then

f(1) = 2 and g(1) = 5, then (f + g)(1)= f(1) + g(1) = 2 + 5 ⇒ (1,2 + 5) ∈ f + g

f +g = {( 1, 2+5), (2, 3+6), (3, 4+7)}

Þ f +g = {(1, 7), (2, 9), (3, 11)}

Subtraction of a real function from another:

Let f : X àR and g : X à R be any two real functions, where X ⊂ R. Then we define (f - g) : XàR by (f-g) (a) = f(a) -g(a), for all a Î X.

Example: If f = {(1,2), (2,3), (3,4)} and g = {(1,5), (2,6), (3,7)} then (f -g)(1)= f(1)-g(1) = 2-5

⇒ (1,2 - 5) ∈ f - g

f -g = {( 1,2-5). (2,3-6), (3,4-7)}

Þ f -g = {( 1, -3). (2, -3), (3, -3)}

Multiplication of a real function by a scalar (real number):

Let f : XàR be a real valued function and k be a real number. Then the product kf : XàR is defined by (kf) (a) = kf (a), a Î X.

Example: If f = {(1,2), (2,3), (3,4)} and k = 2

f(1) = 2, then 2f (1)= 2 x 2 ⇒ (1,2 x 5) ∈ 2f

2f = {(1, 2×2), (2, 2×3), (3, 2×4)} Þ 2f = {(1,4), (2,6), (3,8)}

Multiplication of two real functions:

The product (or multiplication) of two real functions f : XàR and g : XàR is a function fg : XàR defined by (fg) (a) = f(a) g(a), for all a ÎX.

If f = {(1,2), (2,3), (3,4)} and g = {(1,5), (2,6), (3,7)}

f(1) = 2 and g(1) = 5, then (f  x g)(1)= f(1) x g(1) = 2 x 5 ⇒ (1,2 x 5) ∈ f x g

f g = {( 1,2×5). (2,3×6), (3,4×7)}

⇒ f g = {( 1, 10), (2, 18), (3, 28)}

Quotient of two real functions

Let f : X àR and g : X à R be any two real functions, where X ⊂ R. Then we define (f/g) : XàR by (f/g) (a) = f(a) /g(a), provided g(a) ¹0, for all a Î X.

If f = {(1,2), (2,3), (3,4)} and g = {(1,5), (2,6), (3,7)}

f(1) = 2 and g(1) = 5, then (f /g)(1)= f(1)/g(1) = 2/5 ⇒ (1,2 / 5) ∈ f / g

f /g = {( 1,2/5), (2,3/6), (3,4/7)}

⇒ f /g = {( 1, 2/5), (2, ½), (3, 4/7)}