● 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.

● Let `R` be the set of real numbers.

● Define the real valued function `f : R → R` by `y = f(x) = x` for each `x ∈ R.` Such a function is called the identity function.

● Here the domain and range of `f` are `R.` The graph is a straight line as shown in Fig . It passes through the origin.

● Define the function `f: R → R` by `y = f (x) = c, x ∈ R` where `c` is a constant and each `x ∈ R.`

● Here domain of `f` is `R` and its range is `{c}.`

● The graph is a line parallel to `x`-axis. For example, if `f(x)=3` for each `x∈R,` then its graph will be a line as shown in the Fig .

● A function `f : R→R` is said to be polynomial function if for each `x` in `R, `

● `y = f (x) = a_0 + a_1x + a_2x^2 + ...+ a_n x^n,` where `n` is a non-negative integer and `a_0, a_1, a_2,...,a_n ∈R.`

●The functions defined by `f(x) = x^3 – x^2 + 2,` and `g(x) = x^4 + 2 x` are some examples of polynomial functions,
whereas the function `h` defined by `h(x) = x^(2/3) +2x` is not a polynomial function.

● Rational functions are functions of the type `(f( x))/ (g (x))` ,

where `f(x)` and `g(x)` are polynomial functions of `x` defined in a domain, where `g(x) ≠ 0.`

● The function `f: R→R` defined by `f(x) = |x|` for each `x ∈R` is called modulus function.

● For each non-negative value of `x, f(x)` is equal to `x.` But for negative values of `x`, the value of `f(x)` is the negative of the value of `x,` i.e.,

`f(x)= { tt (( x , if , x >= 0),(-x, if, x< 0))`

The graph of the modulus function is given in Fig .

● Signum function The function `f:R→R` defined by


`f(x)= { tt (( 1, if , x > 0),(0, if, x=0),( -1, if , x < 0))`

is called the signum function.

● The domain of the signum function is R and the range is the set `{–1, 0, 1}`.

● The graph of the signum function is given by the Fig .

● The function `f: R → R` defined by

● `f(x) = [x], x ∈R` assumes the value of the greatest integer, less than or equal to `x.` Such a function is called the greatest integer function.

From the definition of `[x],` we can see that

`tt (( [x]= -1, "for" \ \ –1 ≤ x < 0 ),([x]=0 , "for" \ \ 0 ≤ x < 1 ),( [x] = 1 , "for" \ \ 1 ≤ x < 2 ),([x] = 2, "for" \ \ 2 ≤ x < 3))`

and so on.
The graph of the function is shown in Fig .

`(i) "Addition of two real functions"` Let `f : X → R` and `g : X → R` be any two real functions, where `X ⊂ R.`

Then, we define `(f + g): X → R` by
●`(f + g) (x) = f (x) + g (x),` for all `x ∈ X.`

`(ii) "Subtraction of a real function from another"` Let `f :X → R` and `g:X → R` be any two real functions, where `X ⊂ R.`

Then, we define `(f – g) : X→R` by
● `(f–g) (x) = f(x) –g(x),` for all `x ∈ X.`

`(iii) "Multiplication by a scalar"` Let `f : X→R` be a real valued function and `α` be a scalar. Here by scalar, we mean a real number.
Then the product `α f` is a function from `X` to `R` defined by
● `(α f ) (x) = α f (x), x ∈X.`

`(iv) "Multiplication of two real functions"` The product (or multiplication) of two real functions `f:X→R` and `g:X→R` is a function
● `fg:X→R` defined by `(fg) (x) = f(x) g(x),` for all `x ∈ X.`
● This is also called `"pointwise multiplication."`

`(v) "Quotient of two real functions"` Let `f` and `g` be two real functions defined from `X→R` where `X ⊂R.`
The quotient of `f` by `g` denoted by `f/g` is a function defined by ,
● `(f/g)(x)=(f(x))/(g(x))` provided `g(x) ≠ 0, x ∈ X`