- A function `f : X → Y` is defined to be one-one (or injective), if the images of distinct elements of X under f are distinct,
i.e., for every `x_1, x_2 ∈ X, \ \ f (x_1) = f (x_2)` implies `x_1 = x_2`. Otherwise, f is called many-one.

- The function `f_1` and `f_4` in Fig 1 (i) and (iv) are one-one and the function `f_2` and `f_3` in Fig 1 (ii) and (iii) are many-one.

- A function `f : X → Y` is said to be onto (or surjective), if every element of `Y` is the image of some element of X under f, i.e., for every `y ∈ Y`, there exists an element `x` in `X` such that `f (x) = y`.

- A function `f : X → Y` is said to be into , if every element of `Y` is not the image of some element of X under f, .


- Remark `f : X → Y` is onto if and only if Range of `f = Y .`

- The function `f_3` and `f_4` in Fig 1 (iii), (iv) are onto and the function `f_1` in Fig (i) is not onto (into) as elements, `f` in `X_2` are not the image of any element in `X_1` under `f_1.`

- Let `f : A → B` and `g : B → C` be two functions. Then the composition of `f` and `g`, denoted by `gof`, is defined as the function gof `: A → C` given by gof `(x) = g(f (x)), ∀ x ∈ A`.

E.g. : Let `f : {2, 3, 4, 5} → {3, 4, 5, 9}` and `g : {3, 4, 5, 9} → {7, 11, 15}` be functions defined as `f (2) = 3, f (3) = 4, f(4) = f (5) = 5` and `g (3) = g (4) = 7` and `g (5) = g (9) = 11`. Find `gof`.

Solution: We have `gof (2) = g (f (2)) = g (3) = 7, gof (3) = g (f (3)) = g (4) = 7, gof (4) = g (f (4)) = g (5) = 11` and `gof (5) = g (5) = 11`.

- A function `f : X → Y` is defined to be invertible, if there exists a function `g : Y → X` such that `gof = I_X` and `fog = I_Y`. The function `g` is called the inverse of `f` and is denoted by `f^( –1)`.

- Thus, if f is invertible, then f must be one-one and onto and conversely, if f is one-one and onto, then f must be invertible.

E.g., Let `f : N → Y` be a function defined as `f (x) = 4x + 3`, where, `Y = {y ∈ N : y = 4x + 3` for some `x ∈ N`}. Show that `f` is invertible. Find the inverse
Sol : Consider an arbitrary element `y` of `Y`. By the definition of `Y, y = 4x + 3`, for some x in the domain `N`. This shows that `x=((y-3))/4` Define `g : Y → N` by `g(y) =((y-3))/4`
Now, ` gof (x) = g (f (x)) = g (4x + 3) = ((4x+3-3))/3 =x`
and `fog (y) = f (g (y)) = f((y-3)/4) =(4(y-3))/4 +3 =y`

This shows that `gof = I_N` and `fog = I_Y`, which implies that f is invertible and `g` is the inverse of `f`.

(i) Identity function : R `in` 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 .




(ii) Constant function : 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}`.

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






(iii) Polynomial function : 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 + sqrt 2 x` are some examples of polynomial functions.

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 Modulus function 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 , x ge 0),(-x, x < 0))`

The function `f:R→R` defined by `f(x) = { tt (( 1, if , x > 0) , (0 , if , x=0), ( -1 , if , x < 0))`

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.

[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

- 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`.

- 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`.

- 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`.

- 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.

- 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`