06/05/2025
π©A relation between two sets is a collection of ordered pairs containing one object from each set.
If the object x is from the first set and the object y is from the second set, then the objects are said to be related if the ordered pair (x,y) is in the relation.
Relation ααα΅ α΅α₯α΅α¦α½ α α΅ααα΅ α¨αααα±α α΅ αααα΅ ααα’
αα³α
Suppose there are two sets X = {4, 36, 49, 50} and Y = {1, -2, -6, -7, 7, 6, 2}. A relation that states that "(x, y) is in the relation R if x is a square of y" can be represented using ordered pairs as R = {(4, -2), (4, 2), (36, -6), (36, 6), (49, -7), (49, 7)
π₯ Definition and examples of functions
People also ask
What is the formal definition of a function?
A function is a relation that uniquely associates members of one set with members of another set.
Function ααα΅ α¨ relation α ααα΅ α²αα α¨ x value αα°αα α¨αα α΅αα’
αα³α
(2,4),(3,4),(5,6) is functions
(4,2),(3,5),(4,7) is not functions because 4 is repeated.
π₯Classification of functions (one to one, onto, even and odd) and inverse of functions
π©An injective function (injection) or one-to-one function is a function that maps distinct elements of its domain to distinct elements of its codomain. Formally, it is stated as, if f(x) = f(y) implies x=y, then f is one-to-one mapped, or f is 1-1.
α αα΅ ααα π αα αα΅ α°α π§ββοΈα’
α αα΅ X αα αα΅ Y.
π©Onto function /serjection/is a function f that maps an element x to every element y. That means, for every y, there is an x such that f(x) = y.
Y α«α X αααα₯ α¨αα α΅α α’
π©if f(x) produces y, then putting y into the inverse of f produces the output x. x . A function f that has an inverse is called invertible and the inverse is denoted by fβ1.
π©A function f is even if the following equation holds for all x and βx in the domain of f : f(x)=f(βx)
Geometrically, the graph of an even function is symmetric with respect to the y -axis, meaning that its graph remains unchanged after reflection about the y -axis.
π©A function f is odd if the following equation holds for all x and βx in the domain of f : βf(x)=f(βx)
Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after a rotation of 180β about the origin.
π₯Operations on functions and composition of functions
π©Sum(f + g)(x) = f(x) + g(x)
π©Difference(f - g)(x) = f(x) - g(x)
π©Product(f * g)(x) = f(x) * g(x)
π©Quotient(f / g)(x) = f(x) / g(x)
π©In the composition of (f o g) (x) the domain of function f becomes g(x). The domain is a set of all values which go into the function.
Example: If f(x) = 3x+1 and g(x) = 2x , then f of g of x, f(g(x)) = f(x2) = 3(2x)+1. vist our website https://www.12.ofijan.com/
