Math, asked by Anonymous, 11 months ago

Maths question :

What is function ?

Describe all type of function.

Also tell domain and range of all those function .

Need good and quality answer. No spam and no copied.

If know then only give answer.

Answers

Answered by shadowsabers03

First we have to be familiar with relations.

A relation, say R, from a non-empty set A to another non-empty set B, written as R : A → B is anyways a subset of the cross product of A and B, i..e, A × B.

Similarly, A relation, say R, from B to A, written as R : B → A is a subset of B × A.

We know A × B = {(a, b) : a ∈ A, b ∈ B}. So R is also a set of ordered pairs which are in A × B. (R ⊂ A × B)

The set of first elements in such ordered pairs in R is called domain of R, and that of second elements in the ordered pairs is called range of R.

The set to which relation is defined is called co-domain of R.

In R : A → B, B is the co-domain of R. Co-domain of R is always a superset of the range of R.

The elements in B which are related by those in A in R are called the images of the corresponding elements of A. Also, the elements in A which relate those in B are called the pre-images of the corresponding elements in B.

Means, let (a, b) ∈ R. Here 'b' is the image of 'a' while 'a' is the pre-image of 'b'.

Now let's discuss about functions.

A function from A to B written as f : A → B, is a relation where each and every elements of A has one and only one image in B.

Means, in the function f defined by f : A → B, all elements in A should relate those in B, but only one element in B at one time.

In f, there does not exist at least one element in A having no image in B or more that one image in B. That's all!

f is also a set of ordered pairs, since f is also a relation. In the case of f, the first element of each ordered pair in f should not be repeated, but the second element can be.

In f : A → B, since every elements in A relates B, the set A itself is the domain of f. Range and co-domain of f are the same as those of R.

Usually relations are represented by arrow diagrams, while functions are represented by graphs.

All functions are relations but all relations are not functions!

Some types of functions are given below:

(1) Identity Function:

A function f : R → R defined by f = {(x, y) : x, y ∈ R, y = x} is called identity function. Here each ordered pair element in f is in the form (a, a), ∀a ∈ R.

[Note that R here (and there in continuation) indicates the set of all real numbers, not any relation.]

Here the domain and range of f is R.

(2) Constant Function:

A function f : R → R defined by f = {(x , y) : x, y ∈ R, y = c} where c is any constant, is called a constant function. Here each element in f is in the form (x, c), where x is a variable and c is a constant and x, c ∈ R. [∵ f : R → R]

Here the domain is R and the range is {c}.

(3) Polynomial function:

A function f : R → R is said to be polynomial function if,

$f=\left\{(x,\ y):x,\ y\in\mathbb{R},\ y=\displaystyle\sum_{k=0}^{n}a_{_k}x^{k}\right\}$

Here the sum is just an n'th degree polynomial, where n is a non-negative integer and the coefficient of each term belongs to R.

$\displaystyle\sum_{k=0}^{n}a_{_k}x^k\ =\ a_{_0}+a_{_1}x+a_{_2}x^2+a_{_3}x^3+ \dots +a_{_n}x^n\\ \\ \\ n\in\mathbb{Z},\ \ n\geq0,\ \ \{a_{_0},\ a_{_1},\ a_{_2},\ a_{_3}, \dots,\ a_{_n}\}\subseteq \mathbb{R}$

Here the domain is R and the range is $\displaystyle\left\{\sum_{k=1}^{n}a_{_k}x^k\right\}\ \subseteq\ \mathbb{R}.$

(4) Rational function:

A function f : X → R where X ⊆ R defined by f = {(x, y) : x, y ∈ R, y = p(x) / q(x)}, where p(x) and q(x) are polynomial functions having same domain of f, X, and q(x) ≠ 0, is a rational function.

Here the domain is X ⊆ R and the range is {p(x) / q(x)}.

(5) Modulus Function:

A function f : R → R defined by f = {(x, y) : x, y ∈ R, y = |x|} is called modulus function. Each element in f is in the form (x, |x|), i.e.,

$\displaystyle\left\{\begin{array}{lr}\!\!(x,\ x),&x\geq 0\\ \!\!(x,\ -x),&x<0\end{array}$

The domain is R and the range is the interval [0, ∞), which is the set of all non-negative real numbers.

(6) Signum Function:

The function f : R → R is said to be signum function if,

$\displaystyle f=\left\{(x,\ y):x,\ y\in\mathbb{R},\ y=\left\{\begin{array}{rr}1,\ \ \ &x>0\\ 0,\ \ \ &x=0\\ -1,\ &x<0\end{array}\ \ \right\}$

The domain is R and the range is {-1, 0, 1}.

(7) Greatest Integer Function:

The function f : R → R defined by f = {(x, y) : x, y ∈ R, y = [x] }, where [x] represents the greatest integer less than or equal to x, is called greatest integer function.

[x], in this case, is also called the floor function of x, and can also be represented as $\lfloor x\rfloor$ instead of [x].