● We study a special type of relation called function. It is one of the most important concepts in mathematics.
● We can, visualise a function as a rule, which produces new elements out of some given elements. There are many terms such as ‘map’ or ‘mapping’ used to denote a function.
`\color{blue} ☛\color{blue} ul("Function") `
A relation `f` from a set `A` to a set `B` is said to be a function if every element of set `A` has one and only one image in set `B`.
`\color{green} ✍️` In other words, a function `f` is a relation from a non-empty set `A` to a non-empty set `B` such that the domain of `f` is `A` and no two distinct ordered pairs in `f` have the same first element.
`\color{green} ✍️ ` If `f` is a function from `A` to `B` and `(a, b) ∈ f,` then `f (a) = b,` where `b` is called the image of a under `f` and `a` is called the preimage of `b` under `f.`
`\color{green} ✍️ ` The function `f` from `A` to `B` is denoted by `f: A -> B.`
`\color{blue} ☛ color{blue} ul ("Real valued function")`
A function which has either `R` or one of its subsets as its range is called a real valued function. Further, if its domain is also either `R` or a subset of `R`, it is called a real function.
● We study a special type of relation called function. It is one of the most important concepts in mathematics.
● We can, visualise a function as a rule, which produces new elements out of some given elements. There are many terms such as ‘map’ or ‘mapping’ used to denote a function.
`\color{blue} ☛\color{blue} ul("Function") `
A relation `f` from a set `A` to a set `B` is said to be a function if every element of set `A` has one and only one image in set `B`.
`\color{green} ✍️` In other words, a function `f` is a relation from a non-empty set `A` to a non-empty set `B` such that the domain of `f` is `A` and no two distinct ordered pairs in `f` have the same first element.
`\color{green} ✍️ ` If `f` is a function from `A` to `B` and `(a, b) ∈ f,` then `f (a) = b,` where `b` is called the image of a under `f` and `a` is called the preimage of `b` under `f.`
`\color{green} ✍️ ` The function `f` from `A` to `B` is denoted by `f: A -> B.`
`\color{blue} ☛ color{blue} ul ("Real valued function")`
A function which has either `R` or one of its subsets as its range is called a real valued function. Further, if its domain is also either `R` or a subset of `R`, it is called a real function.