Hello! RELATION AND FUNCTION Hi! CONCEPT OF RELATION 1. A relation between two sets is a collection of ordered pairs containing one object from each set. If the object 𝑥𝑥 is from the first set and the object 𝑦𝑦 is from the second set, then the objects are said to be related if the ordered pair (𝑥, 𝑦)(𝑥, 𝑦) is in the relation. CONCEPT OF RELATION 2. A function is a type of relation. But, a relation is allowed to have the object 𝑥𝑥 in the first set to be related to more than one object in the second set. So a relation may not be represented by a function machine, because, given the object 𝑥𝑥 to the input of the machine, the machine couldn't spit out a unique output object that is paired to 𝑥𝑥. CONCEPT OF FUNCTION 1. Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet CONCEPT OF FUNCTION 2. If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x. 3. This relationship is commonly symbolized as y = f(x). In addition to f(x), other abbreviated symbols such as g(x) and P(x) are often used to represent functions of the independent variable x, especially when the nature of the function is unknown or unspecified. CONCEPT OF NOTATION 1. Notation can be defined as a system of symbols or signs that denote elements such as phrases, numbers, words etc. 2. Therefore, function notation is a way in which a function can be represented using symbols and signs. Function notation is a simpler method of describing a function without a lengthy written explanation. CONCEPT OF NOTATION 3. The most frequently used function notation is 𝑓(𝑥) which is read as “𝑓” of “𝑥”. In this case, the letter 𝑥, placed within the parentheses and the entire symbol 𝑓(𝑥), stand for the domain set and range set respectively. 4. Although 𝑓 is the most popular letter used when writing function notation, any other letter of the alphabet can also be used either in upper or lower case. CONCEPT OF DOMAIN AND RANGE 5. The domain of a function 𝑓(𝑥) is the set of all values for which the function is defined, and the range of the function is the set of all values that ff takes. TYPES OF FUNCTION One to one function(Injective) A function is called one to one if for all elements a and b in A, if f(a) = f(b),then it must be the case that a = b. It never maps distinct elements of its domain to the same element of its co-domain. Onto Function (surjective) If every element b in B has a corresponding element a in A such that f(a) = b. It is not required that a is unique; The function f may map one or more elements of A to the same element of B. One to one correspondence function(Bijective/Invertible) A function is Bijective function if it is both one to one and onto function. Inverse Functions: Bijection function are also known as invertible function because they have inverse function property. The inverse of bijection 𝑓 is denoted as 𝑓 −1 . It is a function which assigns to b, a unique element a such that 𝑓(𝑎) = 𝑏. hence 𝑓 −1 (𝑏) = 𝑎. Thanks CREDITS: This presentation template was created by Slidesgo, including icons by Flaticon and infographics & images by Freepik.
