Types of Functions

A Function from set M to set N is a binary relation or a rule which links or plots or pictures each and every component of set M with a component in set N. The purpose of this chapter is to make you learn about various types of functions so that you can become acquainted with the types. You will also come to know that each type has its own individual graphs. Examples of the different types of functions are shown below.

The denotation of function in Mathematics

A function from set M to set N is denoted by:

F: M→N

We chiefly use F, G, H to denote a function

We can also denote a Mathematical class of any function using the following method:

• Tabulation Method

• Graph method

• Arrow Diagram method

A function is defined as a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A and B be two non-empty sets, mapping from A to B will be a function only when every element in set A has one end and only one image in set B. It can be defined that a function is a special relation which maps each element of set A with one and only one element of set B. Both sets A and B must not be empty. A function will define a particular output for a particular input. Therefore, f: A → B is a function such that for a ∈ A there is a unique element b ∈ B such that (a, b) ∈ f. For every Mathematical expression if it has an input value and a resulting answer can be presented as a function. 

Types of Functions in Mathematics with Examples

Types of functions are generally classified into four different types: Based on Elements, Based on Equation, Based on Range, and Based on Domain.

1. Based on Elements:

• One One Function

• Many One Function

• Onto Function

• One One and Onto Function

• Into Function

• Constant Function

2. Based on Equation:

• Identity Function

• Linear Function

• Quadratic Function

• Cubic Function

• Polynomial Functions

3. Based on the Range:

• Modulus Function

• Rational Function

• Signum Function

• Even and Odd Functions

• Periodic Functions

• Greatest Integer Function

• Inverse Function

• Composite Functions

4. Based on the Domain:

• Algebraic Functions

• Trigonometric Functions

• Logarithmic Functions

Types of Function - Based on Elements

1. One-To-One Function.

A Mathematical function is said to be a One-To-One Function if every component of the Domain function possesses its own and unique component in Range of the Function. That being said, a function from set M to set N is considered a One-To-One Function if no two or more elements of set M have the same components mapped or imaged in set N. Also, that no two or more components refined through the function provide the similar output.

For Example:

When f: M→N is described by formula y= f (x) = x³, the function “f” is stated to be a One-To-One function since a cube of different numbers is always different itself.

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2. Onto Function.

A Function is Onto Function if two or more components in its Domain have the same component in its Range.

For Example:

If set M= {M, N, O} and set N= {1,2}

And “f” is a function by which f: M→N is described by:

Then the function “f” is regarded as Onto Function.

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3. Into Functions

A function is said to be an Into function in which there is an element of co-domain Y and does not have a pre-image in domain X.

Example:

Take into account, P = {P, Q, R} 

Q = {1, 2, 3, 4}   and f: P→ Q in a way

f = {P,1, Q,2, R,3}

In the function f, the range i.e., {1, 2, 3} ≠ co-domain of Y i.e., {1, 2, 3, 4} 

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4. One - One Into Functions

The function f is said to be one-one into a function if there exists different components of X and have distinctive unique images of Y.

Example: Prove one-one into function from below set

X = P,Q,R

Y = [1, 2, 3, and 4} and f: X → Y in a way

f = {P,1, (Q,3, R,4}

X = P,Q,R 

Y = [1, 2, 3, and 4} and f: X → Y in a way

f = {P,1, (Q,3, R,4}

Thus, function f is a one-one into function

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5. Many-One Functions

The function f is many-one functions if two or more different elements in X have the same image in Y.

Example: Prove many-one function

Taken, X = 1,2,3,4,5

Y = XYZ

X,Y,Z and f: X → Y

Thus and thus f = {1,X,2,X,3,X,4,Y,5,Z} 

Hence, function f is a many-one function

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6. Many-One Into Functions

 The function f is a many-one function only if it is—both many ones and into a function.

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7. Many-One Onto Functions

The function f is many-one onto function only if is –both many ones and onto.

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8. Constant Function

A constant function is one of the important forms of a many to one function. In this domain every element has a single image. 

The constant function is in the form of 

f(x) = K, where K is a real number.

Types of Function - Based on Equation

Identify Function: The function that has the same domain and range.

Constant Function: The polynomial function of degree zero.

Linear Function: The polynomial function of degree one.

Quadratic Function: The polynomial function of degree two.

Cubic Function: The polynomial function of degree three.

Types of Functions - Based on Range

Modulus Function

The modulus function is the type of function that gives the absolute value of the function, irrespective of the sign of the input domain value. 

The modulus function is defined as f(x) = |x|. 

The input value of 'x' can be a negative or a positive expression.

Rational Function

A Rational Function is the type of function that is composed of two functions and expressed in the form of a fraction X.

A rational function is of the form 

f(x)/g(x), and g(x) ≠ 0. 

Signum Function

The signum function is the type of function that helps to know the sign of the function and does not give the numeric value or any other values for the range.

Even and Odd Function

The even and odd function are the type of functions that are based on the relationship between the input and the output values of the function.

Periodic Function

The function is said to be a periodic function if the same range appears for different domain values and in a sequential manner. 

Inverse Function

The inverse of a function is the type of function in which the domain and range of the given function is reverted as the range and domain of the inverse function.

The inverse function f(x) is denoted by f-1(x).  

Greatest Integer Function

The greatest integer function is the type of function that rounds up the number to the nearest integer less than or equal to the given number.

The greatest integer function is represented as

 f(x) = ⌊x⌋. 

Composite Function

The composite function is the type of function that is made of two functions that have the range of one function forming the domain for another function.

Types of Functions - Based on Domain

Algebraic Function

An algebraic function is the type of function that is helpful to define the various operations of algebra. This function has a variable, coefficient, constant term, and various arithmetic operators such as addition, subtraction, multiplication, division.

Trigonometric Functions

The trigonometric function is the type of function that has a domain and range similar to any other function. The 6 trigonometric functions are :

f(θ) = sinθ, f(θ) = tanθ, f(θ) = cosθ, f(θ) = secθ, f(θ) = cosecθ.

Logarithmic Functions

Logarithmic functions are the type of function that is derived from the exponential functions. The logarithmic functions are considered to be the inverse of exponential functions.

Solved Example of Functions

1. Find the inverse function of the function f(x) = 5x + 4.

Solution: The given function is f(x) = 6x + 4

It is rewritten as y = 6x + 4 and then simplified to find the value of x.

y = 6x + 4

y - 4 = 6x

x = (y - 4)/6

f-1(x) = (x - 4)/6

Ans: So the answer of this inverse function is f-1(x) = (x - 4)/6

2. For the given functions f(x) = 3x + 2 and g(x) = 2x - 1, find the value of fog(x).

Solution: The given two functions are f(x) = 3x + 2 and g(x) = 2x - 1. 

The function fog(x) is to be found.

fog(x) = f(g(x))

= f(2x-1)

= 3(2x - 1) + 2

= 6x - 3 + 2

= 6x - 1

Ans: Therefore fog(x) = 6x - 1

Representation of Functions

The functions can be represented in three ways: Venn diagrams, graphical formats, and roster forms. 

Venn Diagram: The Venn diagram is one of the important formats for representing the function. The Venn diagrams are generally presented as two circles with arrows connecting the element in each of the circles. 

Graphical Form: It is said that every function is easy to understand if they are represented in the graphical form with the help of the coordinate axes. The function in graphical form, helps to understand the changing behavior of the functions if the function is increasing or decreasing.

Roster Form: Roster form is a set of a simple Mathematical representation of the set in Mathematical form. The domain and range of the function in Roster form are represented in flower brackets with the first element of a pair representing the domain and the second element representing the range.

Practice Problems

Practice Problem-1:

Alex leaves his apartment at 5:50 a.m. and goes for a 9-mile jog. He returns at 7:08 a.m. to answer the following questions, assuming Alex runs at a persistent pace.

Report the distance D (in miles) Alex jogs as a linear function of his run time ‘t’ (in minutes).

Draw a graph of D

Simplify the sense of the slope.

Solution-1:

(i) At time t=0, Alex is at his apartment, thus, D (0) =0

At time t= 78 minutes, Alex completed running 9 mi, thus, D (78) =9.

The slope of the linear function comes about as:-

m=9−0 / 78−0= 3 / 26

The y-intercept is (0, 0), thus, the linear equation for this function is

D (t) =3/26 t

(ii) Now, to graph D, execute the fact that the graph cross over the origin and has slope m=3/26

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(iii) The slope m= 3/26 ≈ 0.115 reports the distance (in miles) Alex runs per minute or his average velocity.

Fun Facts

• As per Math processing, there are an infinite number of functions, much more than what you learned in this chapter

• Different Mathematical functions can make us protected in life as being misemployed, deceived or exploited.