# Math Function Types and Properties

## Definition of a Function

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 xâˆˆD a unique element yâˆˆC

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:Dâ†’ C.

The set f(D)={f(x):xâˆˆD}is called the **range** of the function f:Dâ†’ 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.

## What is a Real Function?

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.

### What is one-to-one function?

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

xâ‰ y => f(x)â‰ f(y) âˆ€ x,yâˆˆD.

Or f(x)=f(y)=>x=y âˆ€ x,yâˆˆD.

### What is one-to-one onto function ?

A function f:Dâ†’C is said to be one-one and onto if f is both one-one and onto

### What is an Identity Function ?

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

f:Râ†’R

Such that y = f(x) = x âˆ€ xâˆˆR

Domain of f = R

Codomain of f = R

Range of f = r

Also x â‰ y

=>f(x) â‰ f(y) âˆ€ xâˆˆ

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 câˆˆR be a fixed real number.

Define f:Râ†’R

Such that y=f(x)=c âˆ€ xâˆˆ

Domain of f = R

CoDomain of f = R

Range of f={c}

Also we have xâ‰ y still f(x) f(y)

F is called the constant function

F is neither one-one nor onto

### What is a Polynomial Function

A function f: R â†’R is sad to be polynomial function if for each xâˆˆR we have

y=f(x) = a_{0} + a_{1} x + a_{2} x^{2} + â‹¯ + a_{n} x^{n}

Where n is non-negative integer and a_{0},a_{1},â€¦â€¦..,a_{n} âˆˆR.

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

If a_{1} = a_{2} =a_{3} = â‹¯= 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 â†’R.

Such that y = f(x) = x^{2} âˆ€ x âˆˆ R

Domain of f = R.

Range of f = {x âˆˆ R âˆ¶ x â‰¥ 0}

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

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

f is neither one-one nor onto.

### What are Rational Functions?

Define f: R â†’R.

Such that f(x)=p(x)/q(x) ,q(x) â‰ 0,x âˆˆ R

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

f(x) is called a rational function.

Consider f(x) = 1/x, x â‰ 0, x âˆˆ R

Domain of f = R - {0}

Range of f = R - {0}

f:R - {0} â†’ R - {0} is both one-one and onto.

### What is a Modulus Function ?

Modulus function

Define f: R â†’R.

Such that

f is known as modulus function

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

Domain of f = R

Range of f={x âˆˆRâˆ¶xâ‰¥0}

f is neither one-one nor onto.

The following properties of modulus function cab be easily verified:

- (i) |xy|=|x||y| âˆ€ x,y âˆˆR
- (ii) |x|
^{2}= x^{2}âˆ€ x âˆˆ R - (iii) |x| â‰¤c if and only if-câ‰¤xâ‰¤c
- (iv) -|x| â‰¤x â‰¤|x| âˆ€ xâˆˆ R
- (v) |x+y| â‰¤|x|+|y| âˆ€ x âˆˆ R
- (vi) |(|x|-|y|)| â‰¤|x-y| âˆ€ x, y âˆˆ R
- (vii) |x-y| â‰¤|x|+|y| âˆ€ x,yâˆˆ R

The interval (a+Ïµ , a-Ïµ) is nothing but ={x âˆˆRâˆ¶|x-a| < Îµ}

### What is a Signum Function

f: R â†’ 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.

### Define greatest integer Function

f: R â†’R.

Such that f(x) = [x] âˆ€ x âˆˆ 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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