Let D and C be two non-empty sets. A function f from set D to set C is a rule which associates to each element x∈D a unique element y∈C

The element y, written as y=f(x) is called **image** of x and x is called **preimage** of y .the set D is called the **domain** and C the **co-domain** of function f which is denoted by f:D→ C.

The set f(D)={f(x):x∈D}is called the **range** of the function f:D→ C.

Image of an element is unique in the sense that no element of D can have more than one image. However preimage may not be unique i.e., an element of C can have more than one preimage.

Every element in D has as image in C but every element in C may not have preimage in D. i.e., usually f(D) is proper subset of C.

A function which has either R or one of its subsets as its range is called a real values function. Further ,if its domain is also either R or a subset of R ,It is called a real function.

A function f: D-> C is called one-to-one if distinct elements of D have distinct images in C, i.e.

x≠y => f(x)≠f(y) ∀ x,y∈D.

Or f(x)=f(y)=>x=y ∀ x,y∈D.

A function f:D→C is said to be one-one and onto if f is both one-one and onto

Identify Function: Let R be the set of real numbers. Define the real Function

f:R→R

Such that y = f(x) = x ∀ x∈R

Domain of f = R

Codomain of f = R

Range of f = r

Also x ≠ y

=>f(x) ≠f(y) ∀ x∈

f is called identify function.

F is both one-one and onto.

**What is a Constant Function** Let R be the set of real numbers and c∈R be a fixed real number.

Define f:R→R

Such that y=f(x)=c ∀ x∈

Domain of f = R

CoDomain of f = R

Range of f={c}

Also we have x≠y still f(x) f(y)

F is called the constant function

F is neither one-one nor onto

A function f: R →R is sad to be polynomial function if for each x∈R we have

y=f(x) = a_{0} + a_{1} x + a_{2} x^{2} + ⋯ + a_{n} x^{n}

Where n is non-negative integer and a_{0},a_{1},……..,a_{n} ∈R.

If a_{0} = a_{2} = a_{3} = ⋯= a_{n} = 0 and a_{1} = 1 the polynomial function become identity function.

If a_{1} = a_{2} =a_{3} = ⋯= a_{n} = 0 the polynomial function become constant function.

f(x) = x^{2} + 2x = 1 is a polynomial function.

f(x) = x^{3/2} + 4x is not a polynomial function.

Consider the function

f: R →R.

Such that y = f(x) = x^{2} ∀ x ∈ R

Domain of f = R.

Range of f = {x ∈ R ∶ x ≥ 0}

f(-1) = (-1)^{2}

= 1 = 1^{2} = f(1)

f is neither one-one nor onto.

Define f: R →R.

Such that f(x)=p(x)/q(x) ,q(x) ≠ 0,x ∈ R

Where p(x) and q(x) are polynomials in x.

f(x) is called a rational function.

Consider f(x) = 1/x, x ≠ 0, x ∈ R

Domain of f = R - {0}

Range of f = R - {0}

f:R - {0} → R - {0} is both one-one and onto.

Modulus function

Define f: R →R.

Such that

f is known as modulus function

f(-1)=|-1|=1=|1|=f(1)

Domain of f = R

Range of f={x ∈R∶x≥0}

f is neither one-one nor onto.

The following properties of modulus function cab be easily verified:

- (i) |xy|=|x||y| ∀ x,y ∈R
- (ii) |x|
^{2}= x^{2}∀ x ∈ R - (iii) |x| ≤c if and only if-c≤x≤c
- (iv) -|x| ≤x ≤|x| ∀ x∈ R
- (v) |x+y| ≤|x|+|y| ∀ x ∈ R
- (vi) |(|x|-|y|)| ≤|x-y| ∀ x, y ∈ R
- (vii) |x-y| ≤|x|+|y| ∀ x,y∈ R

The interval (a+ϵ , a-ϵ) is nothing but ={x ∈R∶|x-a| < ε}

f: R → R.

Such that

f is called signum function.

Domain of f = R.

Range of f={-1,1,0}

F is neither one-one nor onto.

f: R →R.

Such that f(x) = [x] ∀ x ∈ R

=Greatest integer not greater than x

([1.1]=1&[0.9]=0&[1]=1

[-1.1]=-2&[-0.9]=-1&[-1]=-1)

Domain of f = R

Range of f = Z

f is neither one-one nor onto.

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