Classification of functions: Mathematically, functions are defined as relations where every input has a particular output. Many commonly used formulas are functions. For example, the perimeter of a circle, \(P=2πr\) is a function of the radius, and the area of a triangle, \(A = \frac{1}{2}bh\) is a function of the base and height.

Functions are omnipresent in Mathematics and play a significant role in establishing physical relationships in Science. There are different types of functions based on several factors such as the domain, range, relation, and equations. In this article, let us learn about the different types and classifications of functions.

Classification of Functions Based on Relation

Injective or One-One Functions

This function maps the distinct elements of the domain to the distinct elements of the co-domain.

Definition: A function where every element in the first set has a unique image in the second set is said to be injective. In simpler words, for \(f:A \to B\) to be injective, each element of set \(A\) has a unique image in set \(B.\)

For every \({a_1},{a_2} \in A,f\left( {{a_1}} \right) = f\left( {{a_2}} \right) \Rightarrow {a_1} = {a_2}.\)

Surjective or Onto Functions

This function maps all the elements of the co-domain to at least one element of the domain.

Definition: A function where each image in \(B\) has at least one pre-image in \(A.\)

Note that for Surjective or onto functions, the range of the function is the co-domain itself.

For every \(b \in B,\) there exists an element in \(a\) in \(A\) such that \(f(a) = b.\)

Bijective or One-One and Onto Functions

If a function \(f:A \to B\) is both one-one (injective) and onto (surjective), it is said to be bijective.

Definition: A function where every element in a set \(A\) has a distinct image in set \(B,\) and there are no unmapped elements in \(B.\)

Many-One Functions

These functions give the same output for more than one input.

Definition: A function where more than one element in the first set have the same image in the second set is called a many-one function. In simpler words, for \(f:A \to B\) to be many-one, at least two elements of set \(A\) has the same image in set \(B.\)

Into Functions

These functions are those that have unmapped elements in the second set.

Definition: A function where at least one image in set \(B\) does not have a pre-image in set \(A.\)

Constant Functions

As the name suggests, the output of the function is always a constant, immaterial of the input. It is represented as

\(f\left( x \right) = c\)

Where, \(c\) is a constant.

A constant function can mathematically be expressed as \(f:A→A\)

Classification of Functions Based on Range

Modulus Functions

A modulus function returns the absolute value of the input. It is represented as

\(y = \left| x \right|\)

The graph of a modulus function is shown below.

Expression	\(f(x) = \left\{ {\begin{array}{*{20}{c}} {x,\quad x \ge 0}\\ { – x,\quad x < 0} \end{array}} \right.\)
Domain	\(\mathbb{R}\)
Range	\([0,\infty )\)

Inverse Functions

Any function that undoes the operation of \(f\) is called an inverse function. This is achieved by interchanging the dependent and independent variables in a function. For a bijective function \(f:A \to B,\) its inverse is \({f^{ – 1}}:B \to A.\)

Expression	\({f^{ – 1}}\left( x \right)\)
Domain of the inverse function	Range of the actual function
Range of the inverse function	Domain of the actual function

Rational Functions

A ratio of two polynomial functions is called a rational function.

Expression	\(f(x) = \frac{{A(x)}}{{B(x)}}\) where, \(B(x) \ne 0\)
Domain	\(\mathbb{R}\)
Range	\(\mathbb{R}\)

Even and Odd Functions

	Even Function	Odd Function
Expression	\(f( – x) = f(x)\)	\(f( – x) = – f(x)\)
Domain	\(\mathbb{R}\)	\(\mathbb{R}\)
Range	\(\mathbb{R}\)	\(\mathbb{R}\)
Line of Symmetry	\(y-\)axis	\(x-\)axis
Example	\(f(x) = \cos x\)	\(f(x) = \sin x\)

Signum Functions

This function outputs the sign of the real-value function. It gives an output of \(+1\) for positive inputs and \(-1\) for negative inputs.

Expression	\(f(x) = \left\{ {\begin{array}{*{20}{c}} {1,}&{x > 0} \\ { – 1,}&{x < 0} \\ {0,}&{x = 0} \end{array}} \right.\)
Domain	\(\mathbb{R}\) on \(x-\)axis
Range	\([-1,1]\) on \(y-\)axis

Greatest Integer Function

Also known as the step curve, this function identifies the greatest integer equal to or less than the given number.

Expression	\(f(x) = \left[ x \right]\)
Domain	\(\mathbb{R}\)
Range	\(\mathbb{Z}\)

• If the input is an integer, the function returns the number itself.
• If the input is a non-integer, then the output of the function is the integer that lies just before the input number.

Smallest Integer Function

The output of the smallest integer function is the smallest integer that is bigger than or equal to the input number.

Expression	\(f(x) = \left[ x \right]\)
Domain	\(\mathbb{R}\)
Range	\(\mathbb{Z}\)

• If the input is an integer, the function returns the number itself.
• If the input is a non-integer, then the output of the function is the integer that lies just after the input number.

Composite Function

As the name suggests, this is a combination of two functions. The output of one function acts as the input to the other function.

Expression	\(f(g(x)\) or \((f^\circ g)(x)\)
Domain	Domain of the function \(f\)
Range	Range of the function \(g\)

Periodic Functions

Any function whose output repeats itself in a specific interval is called a periodic function. A function \(f(x)\) in a period \(P\) is said to be periodic if \(f(x + P) = f(x).\) Here, \(P\) is a positive real number.

Practice Exam Questions

Classification of Functions Based on Domain

Exponential Functions

The functions used to calculate the exponential decay or the exponential growth are called exponential functions. They are expressed as

\(f(x) = {a^x}\)

Here, \(x\) is a variable, and \(a\) is a constant.

Graph of \(f(x) = {2^x}\) is shown below.

Trigonometric Functions

Functions that are defined using ratios of the sides of a right-angled triangle are called trigonometric functions. The trigonometric functions and their inverses are listed below.

sine	cosine	tangent	cotangent	secant	cosecant
arcsine	arccosine	arctangent	arccotangent	arcsecant	arccosecant

Note that trigonometric functions are also periodic functions.

The graph of a sine function is shown below.

Logarithmic Functions

The logarithmic functions are the inverse of exponential functions. They are represented as

\(f(x) = {\log _a}x\)

Here, \(a\) is the base of the function and is a positive number.

For the exponential function \(f(x) = {2^x},\) the corresponding logarithmic function is \({f^{ – 1}}(x) = {\log _2}x.\) The graphs are shown below.

Algebraic Functions

Functions that involve only algebraic operations such as addition, subtraction, multiplication, and division are called algebraic functions. It is expressed as,

\(p(x,f(x)) = 0\)

Where, \(y=f(x)\) is a polynomial.

Clasification of Functions Based on Equations

Identity Functions

A function that does not alter the input is called an identity function. In an identity function, the output is the same as the input. It is given by

\(f(x) = x\)

Expression	\(f(x) = x\)
Domain and Range	\(\{ (1,1),(2,2),(3,3) \ldots (n,n)\} \)
Graph	Straight Line in Quadrant \({\text{I}}\) and Quadrant \({\text{III}}\)

Linear Functions

A function that is defined by a first-degree polynomial is called a linear function.

Expression	\(f(x)=ax+b\) Where, \(a\) and \(b\) are real numbers
Domain	\(\mathbb{R}\)
Range	\(\mathbb{R}\)
Graph	Straight Line

The graph of the linear function \(f(x) = 3x + 2\) is shown below.

Quadratic Functions

Functions that have a second-degree polynomial are called quadratic functions. In simpler words, the highest power in the function is \(2.\)

Expression	\(f(x) = a{x^2} + bx + c\) Where, \(a,b\) and \(c\) are real numbers and \(a≠0\)
Domain	\(\mathbb{R}\)
Range	\(\mathbb{R}\)
Graph	Non-linear, Parabolic

The graph of the linear function \(f(x) = {x^2} – x – 2\) is shown below.

Cubic Functions

As the name implies, the highest power in the equation is \(3.\)

Expression	\(f(x) = a{x^3} + b{x^2} + cx + d\) Where, \(a,b,c\) and \(d\) are real numbers and \(a≠0\)
Domain	\(\mathbb{R}\)
Range	\(\mathbb{R}\)

The graph of a cubic function \(f(x) – 4{x^3} + 3{x^2} + 25x + 6\) is shown below.

Polynomial Functions

Any function that can be represented as a polynomial is called a polynomial function. It is represented as

\(f(x) = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + \cdots + {a_n}{x^n}\)

Where, \({a_0},{a_1},{a_2},{a_3}, \ldots {a_n}\) are all real numbers, and \(n\) is a positive integer.

Note that linear, quadratic, and cubic functions are all polynomial functions.

Solved Problems – Classification of Functions

1. Which of the following are functions?

Solution:
a. This is not a function because one element in the domain does not have an image in the co-domain.
b. This is a function.
c. This is not a function because two elements in the domain have more than one image in the co-domain.
d. This is not a function because two images have the same pre-image.
e. This is a function.

2. Which of these are functions? Identify the type of function.

Solution:

1. Quadratic function
2. Linear function
3. Not a function
4. Polynomial function
5. Not a function

3. Find the inverse of \(f(x) = 4x + 5.\)
Solution:
Given: \(f(x) = 4x + 5\)
\( \Rightarrow y = 4x + 5\)
\( \Rightarrow 4x = y – 5\)
\( \Rightarrow x = \frac{{y – 5}}{4}\)
\(\therefore {f^{ – 1}}(x) = \frac{{x – 5}}{4}\)

4. If \(f(x) = 7 – 2x\) and \(g(x) = 5x + 1,\) find \(g(f(x))\)
Solution:
Given:
\(f(x) = 7 – 2x\)
\(g(x) = 5x + 1\)
Then, \(g(f(x)) = 5(7 – 2x) + 1\)
\( \Rightarrow 35 – 10x + 1\)
\(\therefore g(f(x)) = 36 – 10x\)

5. If \(A = \mathbb{R} – \{ 3\} \) and \(B = \mathbb{R} – \{ 1\} ,\) and \(f:A→B\) is defined by \(f(x) = \frac{{x – 2}}{{x – 3}},\) for all \(x \in A,\) show that \(f\) is bijective.
Solution:
We know that bijective functions are one-one and onto

1. Check for one-one:

Let \({x_1},{x_2} \in A,\) such that \(f\left( {{x_1}} \right) = f\left( {{x_2}} \right)\)
Then, we can write,
\(\frac{{{x_1} – 2}}{{{x_1} – 3}} = \frac{{{x_2} – 2}}{{{x_2} – 3}}\)
\(\left( {{x_1} – 2} \right)\left( {{x_2} – 3} \right) = \left( {{x_2} – 2} \right)\left( {{x_1} – 3} \right)\)
\({x_1}{x_2} – 3{x_1} – 2{x_2} + 6 = {x_1}{x_2} – 3{x_2} – 2{x_1} + 6\)
\( – 3{x_1} – 2{x_2} + 6 = – 3{x_2} – 2{x_1} + 6\)
\(3{x_2} – 2{x_2} = 3{x_1} – 2{x_1}\)
\({x_2} = {x_1}\)
Hence, we can say that \(f(x)\) is one-one.

2. Check for onto:

Let \(y \in B\) be any arbitrary element. Then, we can say that,
\(f(x) = y\)
\( \Rightarrow \frac{{x – 2}}{{x – 3}} = y\)
\(x – 2 = y(x – 3)\)
\(x – 2 = xy – 3y\)
\(x – xy = 2 – 3y\)
Solving for \(x,\) we get,
\(x(1 – y) = 2 – 3y\)
\(\therefore x = \frac{{2 – 3y}}{{1 – y}}\)
Substituting for \(x\) in \(f(x),\) we get
\(f\left( {\frac{{2 – 3y}}{{1 – y}}} \right) = \frac{{\left( {\frac{{2 – 3y}}{{1 – y}}} \right) – 2}}{{\left( {\frac{{2 – 3y}}{{1 – y}}} \right) – 3}}\)
\( = \frac{{2 – 3y – 2(1 – y)}}{{2 – 3y – 3(1 – y)}}\)
\( = \frac{{2 – 3y – 2 + 2y}}{{2 – 3y – 3 + 3y}}\)
\( = \frac{{ – y}}{{ – 1}}\)
\(\therefore f\left( {\frac{{2 – 3y}}{{1 – y}}} \right) = y\)
Hence, \(f(x)\) is onto.
Hence proved that \(f(x)\) is bijective.

Summary – Classification of Functions

A function is any relation where every input has a specific output. Functions are classified based on factors such as elements, domain, range, and equations. Based on the domain, the types of functions are algebraic, exponential, logarithmic, and trigonometric. The functions based on the range are modulus, rational, signum, even and odd, periodic, greatest integer, smallest integer, inverse and composite functions.

The functions are classified as identity, linear, quadratic, cubic, and polynomial based on the equation. Lastly, the functions based on the elements are one-one, many-one, onto, into, constant, and one-one and onto functions.

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FAQs on Classification of Functions

Q.1. What is a function?
Ans: A function is a relation where every input has a specific output. It is a rule that defines the relationship between every element in the first to one or more elements in the second set.

Q.2. What is an example of a function?
Ans: A few examples of functions are:
1. \(f(x) = {x^2}\)
2. \(f(y) = \left| {{y^3} + 1} \right|\)
3. \(f(z) = \sin z\)

Q.3. How many classifications of functions are there?
Ans: The various classifications of functions are listed below.

Category of Classification	Types of Functions
Domain	1. Algebraic functions 2. Exponential functions 3. Logarithmic functions 4. Trigonometric functions
Range	1. Composite functions  2. Even and odd functions 3. Greatest integer functions 4. Inverse functions 5. Modulus functions 6. Periodic functions 7. Rational functions 8. Signum function 9. Smallest integer functions
Elements	1. Constant functions 2. Into functions 3. Many-one functions 4. One-one functions 5. One-one and onto functions 6. Onto functions
Equation	1. Cubic functions 2. dentity functions 3. Linear functions 4. Polynomial functions 5. Quadratic functions

Q.4. What is the importance of functions?
Ans: Functions are vital tools in Mathematics that are used to establish and define relationships. It is used to estimate the output based on the various included variables mathematically. They are also the building blocks of machine design, mathematical study and predicting natural disasters.

Q.5. What are domain and range?
Ans: The set of values that are input to the function is called the domain, while the set of values that are the outputs from the function is called the range.

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