Types of Functions

What is a Function in Math?

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

Types of Functions in Mathematics with Examples

As per the characteristics exhibited by a Function, it can be classified into different types such as:-

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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.


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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5. 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]}

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 = [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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Solved Examples

Practice Problem1:

Alex leaves his apartment at 5:50 a.m. and goes for 9-mile jogging. 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.


(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, deceive or exploited

FAQ (Frequently Asked Questions)

Q1. Are there Real-life Applications of Different Function Types?

Yes, apart from mathematics, different types of functions in math are incorporated to compute physical processes like:-

  • The Circumference of a Circle is a function of its diameter represented as C (d)= dπ.

  • The range of a person’s shadow across the floor is a function of their height

  • When riding the bike, your location at that point in time is a function of time.

  • The sum of money you own is a function of the time spent earning it

Q2. What is the Best Use of Functions in Math?

From a more mathematical perspective, here are two functions that give actual, real-world data incorporated by professionals.

Height of a Person - Forensic researchers can identify the height of a person based on the length of their thigh bone. Take a look at one such function: h(f)=3.58 f + 65.21, ± 4.83 cm—Where [f] is the length of the thigh bone.

Position of a person/object- The distance an object/person travels as a function of time is provided by s(t)=14at 3+v0t+y0, where  'a'  is the rate of change in velocity due to gravity (−8.921 m/s 3, or - 34  feet/second),  'v0'  is the initial velocity, and  'y0'  is the initial height.