## Special Types of Functions

• Constant Function
• Identity Function
• Polynomial function
• Linear Function
• Identical Function
• Rational Function
• Algebraic Functions
• Cubic Function
• Modulus Function
• Signum Function
• Greatest Integer Function
• Smallest Integer Function
• Step Function
• Characteristics Function
• Indicator Function
• Fractional Part Function
• Exponential Function
• Logarithmic Function
•  Sinusoidal. Function
• Even and Odd Function
• Periodic Function
• Composite Function

### Polynomial function

A real valued function f : P → P defined by y = f (a)=h0+h1 a+…..+hn an, where n ϵ N, and h0+h1+…..+hn ϵ P, for each a ϵ P, is called polynomial function.

• N = a Natural Number.
• The degree of Polynomial function is the highest power in the expression.
• If the degree is zero, it’s called a constant function.
• If the degree is one, it’s called a linear function. Example: b = a+1.
• Graph type: Always a straight line.

So, a polynomial function can be expressed as :

f(x)=anxn+an−1xn−1+…..+a1x1+a0

The highest power in the expression is known as the degree of the polynomial function. The different types of polynomial functions based on the degree are:

1. The polynomial function is called a Constant function if the degree is zero.
2. The polynomial function is called a Linear if the degree is one.
3. The polynomial function is Quadratic if the degree is two.
4. The polynomial function is Cubic if the degree is three.

### Linear Function

All functions in the form of ax + b where a, bR & a ≠ 0 are called as linear functions. The graph will be a straight line. In other words, a linear polynomial function is a first-degree polynomial where the input needs to be multiplied by m and added to c. It can be expressed by f(x) = mx + c.

For example, f(x) = 2x + 1 at x = 1

f(1) = 2.1 + 1 = 3

f(1) = 3 Another example of linear function is y = x + 3 ### Identical Function

Two functions f and g are said to be identical if

(a) The domain of f = domain of g

(b) The range of f = the Range of g

(c) f(x) = g(x)  xDf & Dg

For example f(x) = x & g(x) =11/x

Solution: f(x) = x is defined for all x

But g(x) = 11/x is not defined of x = 0

Hence it is identical for x R -{0}

All functions in the form of y = ax2 + bx + c where a, b, cR, a ≠ 0 will be known as Quadratic function. The graph will be parabolic. At x=−b±D2, we will get its maximum on minimum value depends on the leading coefficient and that value will be −D4a (where D = Discriminant)

In simpler terms,

A Quadratic polynomial function is a second degree polynomial and it can be expressed as;

F(x) = ax2 + bx + c, and a is not equal to zero.

Where a, b, c are constant and x is a variable.

Example, f(x) = 2x2 + x – 1 at x = 2

If x = 2, f(2) = 2.22 + 2 – 1 = 9 For Example: y = x2 + 1

### Rational Function

These are the real functions of the type f(a)g(a) where f (a) and g (a) are polynomial functions of a defined in a domain, where g(a) ≠ 0.

• For example f : P – {– 6} → P defined by f (a) = f(a+1)g(a+2), aϵP – {–6 }is a rational function.
• Graph type: Asymptotes (the curves touching the axes lines).

### Algebraic Functions

A function that consists of a finite number of terms involving powers and roots of independent variable x and fundamental operations such as addition, subtraction, multiplication, and division is known as an algebraic equation.

For Example,

f(x)=5x3−2x2+3x+6, g(x)=3x+4(x−1)2.

### Cubic Function

A cubic polynomial function is a polynomial of degree three and can be expressed as;

F(x) = ax3 + bx2 + cx + d and a is not equal to zero. In other words, any function in the form of f(x) = ax3 + bx2 + cx + d, where a, b, c, dR & a ≠ 0 For example: y = x3

MODULUS FUNCTION:

A Function f(x) : R à R is said to be Modulus function if y=f(x) = |x| Domain of f = R

Range of f = R+U {0}

Codomain=R If x = -5, then y = f(x) = – (-5) = 5, since x is less than zero

If x = 10, then y = f(x) = 10, since x is greater than zero

If x = 0, then y = f(x) = 0, since x is equal to zero

When x = -3 then y = |-3| = 3

When x = -2 then y = |-2| = 2

When x = -1 then y = |-1| = 0

When x = 0 then y = |0| = 0

When x = 1 then y = |1| = 1

When x = 2 then y = |2| = 2

When x = 3 then y = |3| = 3

Signum Function:

A Function f(x) : R à R is said to be signum function if Domain of f = R

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

Codomain=R If x = -5, then y = f(x) = – 1 , since x is less than zero

If x = 5, then y = f(x) = 1 , since x is greater than zero

If x = 0, then y = f(x) = 0 , since x is equal to zero

Greatest Integer Function:

A  Function f(x) : R à R is said to be greatest integer function if y=f(x)=[x] or

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. Domain of f = R

Range of f = Z

Codomain=R

[1.15] = 1  , [1.9] =1

[4.56567] = 4  , [4.99] = 4

 = 50

[-3.010] = -4

## Greatest Integer Function Properties

•  [x] = x, where x is an integer
• [x + n] = [x] + n, where n  Z
• [-x] = –[x], if x  Z
• [-x] =-[x] – 1, if x  Z
• If [f(x)] ≥ Y, then f(x) ≥ Y

## Ceiling function f(x) = ⌈x⌉ and  floor function f(x)=⌊x ⌋

f(x) = x = Largest Nearest Integer of specified value

f(x) = x = Least Nearest successive Integer of specified value

A  Function f(x) : R à R is said to be Smallest integer function if y=f(x)= x Domain of f = R

Range of f = Z

Codomain=R Properties:

• x + y – 1 ≤ x + yx + y
• x + a = x + a
• x = a; iff x ≤ a < x + 1
• x = a; iff x – 1 < a ≤ x
• a < x iff a < x
• a ≤ x iff x < a

Step Function:

A step function (also called as staircase function) is defined as a piecewise constant function, that has only a finite number of pieces. In other words, a function on the real numbers can be described as a finite linear combination of indicator functions of given intervals.  Domain of f = referred to as the set of input values

Range of f = referred to as the set of output values generated for the domain (input values)

Properties

The important properties of step functions are given below:

• The sum or product of two-step functions is also a step function.
• If a step function is multiplied by a number, then the result produced is again a step function. That indicates the step functions create an algebra over the real numbers
• A step function can take only a finite number of values
• Piecewise linear function is the definite integral of a step function

Indicator function / Characteristic function:

An indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements of the set to zero. The indicator function of a subset A of a set X maps X to the two-element set

{ 0,1}; cA(x)=1 if an element in X belongs to A

and cA(x)=0 if x does not belong to A. It may be denoted as 1A , by IA or by cA

The indicator function of a subset, that is the function

c A : X à {0,1}

which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.

Fractional Part Function:

The fractional part function is the difference between a number and its integer part.

A  Function f(x) : R à R is said to be fractional part function if  y = f(x) = {x} = x- [x]

Domain of a fractional function is all the real numbers except the roots of denominator of the fraction.

Codomain =R

Range = [0,1) {1}=1−=1-1=0.

Properties; Exponential Function:

For a > 1, the Exponential of b to base a is x if y =f(x)=  ax = b. Thus, The  function is known as Exponential function

Exponential function having base 10 is known as a common exponential function.

i.e.,y=f(x)= 10x

Exponential function having base e is known as a Natural exponential function.

y=f(x)= ex

Domain =R

Codomain =R+

Range = R+

Properties:

• The graph passes through the point (0,1).
• The domain is all real numbers
• The range is y>0
• The graph is increasing
• The graph is asymptotic to the x-axis as x approaches negative infinity
• The graph increases without bound as x approaches positive infinity
• The graph is continuous
• The graph is smooth  If a>0, and  b>0, the following hold true for all the real numbers x and y:

• ax ay = ax+y
• ax/ay = ax-y
• (ax)y = axy
• axbx=(ab)x
• (a/b)x= ax/bx
• a0=1
• a-x= 1/ ax

Logarithmic Function:

For a > 1, the logarithm of b to base a is x if ax = b. Thus, loga b = x if ax = b. This function is known as logarithmic function.

logarithm function having base 10 is known as a common logarithm function.

i.e.,y=f(x)= log10 x

logarithm function having base e is known as a Natural logarithm function.

y=f(x)= loge x= lnx

Domain = R+

Codomain =R

Range = R  Properties:

• The graph passes through the point (1,0).
• The domain is all +ve real numbers
• Logap = α, logbp = β and logba = µ, then aα = p, bβ = p and bµ = a
• Logbpq = Logbp + Logbq
• Logbp= ylogbp
• Log(p/q) = logbp – logbq
• When we plot the graph of log functions and move from left to right, the functions show increasing behaviour.
• The graph of log function never cuts the x-axis or y-axis, though it seems to tend toward them.

Sinusoidal Function:

A sinusoidal function is a function using the sine function. The basic form of a sinusoidal function is where A is the amplitude or height of our function, B is the change in period defined by the horizontal shift, and D the vertical shift.​​​​​​​ The sine and cosine functions have several distinct characteristics: ​​​​​​​

Even and Odd Function:

The definition of even and odd functions:

Even function: A function, f(x) is even if f(x) = f(-x)

Example:f(x)=cosx

f(-x)=cos(-x)= cosx=f(x)

so f is even function.

examples of even functions are x4, cot x, y = x2, etc.

Odd Function: A function, f(x) is odd if f(x) = -f(x) For example,

Check if function is even or odd: f (x) = tan x

f(-x)=tan(-x)= -tanx=-f(x)

so f is odd function.

The polynomial function f(x)=x2+x4+x6 is even. The polynomial function f(x)=x+x3+x5 is odd

Periodic Function:

A function y= f(x) is said to be a periodic function if there exists a positive real number P such that f(x + P) = f(x), for all x belongs to real numbers. The least value of the positive real number P is called the fundamental period of a function.

This fundamental period of a function is also called the period of the function, at which the function repeats itself. f(x + P) = f(x) The period of the sine ,cosine function is 2π (units). All the trigonometric functions are periodic functions.

Composite Function:

Let f : A → B and g : B → C be two functions. Then the composition of f and g, denoted by g f, is defined as the function g f : A → C given by g f (x) = g(f (x)), x A. The composite function gof(x) is read as “g of f of x”. If f(x)and g(x) are two functions then fog(x), gof(x), gog(x) and fof(x) are composite functions.

• fog(x) = f(g(x))
• gof(x) = g(f(x))
• gog(x) = g(g(x))
• fof(x) = f(f(x))
• fogoh(x) = f(g(h(x)))
• fofof(x) = f(f(f(x)))

The order of the function is important in a composite function since (fog)(x) is not equal to (gof)(x). 