Types of Functions

The types of functions are defined on the basis of the mapping, degree, and math concepts. The expression used to write the function is the prime defining factor for a function. Along with expression, the relationship between the elements of the domain set and the range set also accounts for the type of function. The classification of functions helps to easily understand and learn the different types of functions.

Every mathematical expression which has an input value and a resulting answer can be conveniently presented as a function provided that every input has only one output. Here we shall learn about the types of functions and their definitions along with their graphs and examples.

The function y = f(x) is classified into different types of functions, based on factors such as how they have been mapped, what is their degree, and what math concepts they belong to. The functions have a domain x value that is referred as input. The domain values (set of x-values) can be a number, angle, decimal, fraction, etc depending on its type. Similarly, the set of y values is the range. The types of functions have been classified into the following four types.

• Based on the mapping
• Based on Degree
• Based on Math Concepts
• Miscellaneous Functions

There are three different forms of representation of functions. The functions need to be represented to showcase the domain values and the range values and the relationship between them. The functions can be represented with the help of algebraic form, graphical formats, and roster forms.

Algebraic Form: A function is usually denoted by an equation y = f(x) which connects the values on the x-axis and the values on the y-axis. Some examples of functions equations are f(x) = x³, f(x) = sin x, etc.

Graphical Form: Functions are easy to understand if they are represented in a graphical form with the help of the coordinate axes. Representing the function in graphical form, helps us to understand the changing behavior of the functions if the function is increasing or decreasing. To understand how the functions are graphed, click here.

Roster Form: Roster notation of a set is a simple mathematical representation of the set in mathematical form. In this notation, a function is represented with a set of points (ordered pairs) on its graph with the first element and second elements of a pair represent the elements of the domain and range respectively. Let us try to understand this with the help of a simple example. For a function of the form f(x) = x², x ∈ N and x ≤ 4, the function is represented as {(1, 1), (2, 4), (3, 9), (4, 16)}. Here the first element is the domain or the x value and the second element is the range or the f(x) value of the function.

The types of functions are classified further to help for easy understanding and learning. The types of functions have been further classified into four different types, and are presented as follows.

These types of functions are classified based on how the elements of the domain and the codomain are mapped. Take a look at the following figure to get an idea about each of these functions.

One One Function

A one-to-one function is defined by f: A → B such that every element of set A is connected to a distinct element in set B. The one-to-one function is also called an injective function. Here every element of the domain has a distinct image or co-domain element for the given function.

Many to One Function

A many to one function is defined by the function f: A → B, such that more than one element of the set A are connected to the same element in the set B. In a many to one function, more than one element has the same image. In a many-to-one function, if there is only a single value in the codomain which is mapped with all the elements of the domain, then it becomes a constant function.

Onto Function

In an onto function, every codomain element is related to the domain element. i.e., In an onto function, no element of codomain is left without being mapped.For a function defined by f: A → B, such that every element in set B has a pre-image in set A. The onto function is also called a subjective function.

One One and Onto Function (Bijection)

A function that is both a one and onto function is called a bijective function. Here every element of the domain is connected to a distinct element in the codomain and every element of the codomain has a pre-image. Also in other words every element of set A is connected to a distinct element in set B, and there is not a single element in set B which has been left out.

Into Function

Into function is exactly opposite in properties to an onto function. Here there are certain elements in the co-domain that do not have any pre-image. The elements in set B are excess and are not connected to any elements in set A.

The algebraic expressions are also functions and are based on the degree of the polynomial. The functions based on equations are classified into the following equations based on the degree of the variable 'x'.

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

Let us understand each of these functions in detail.

Identity Function

The identity function has the same domain and range. The identity function equation is f(x) = x, or y = x. Its ordered pairs are of the form {(1, 1), (2, 2), (3, 3), (4, 4).....(n, n)}.

The graph of the identity function is a straight line that is equally inclined to the coordinate axes and is passing through the origin. The identity function can take both positive and negative values and hence it is present in the first and the third quadrants of the coordinate axis.

Constant Function

A constant function is an important form of a many-to-one function. In a constant function, all the domain elements have a single image. The constant function is of the form f(x) = K, where K is a real number. For the different values of the domain (x value), the same range value of K is obtained for a constant function.

Linear Function

A polynomial function having the first-degree equation is a linear function. The domain and range of a linear function is the set of all real numbers, and it has a straight-line graph. Equations such as y = x + 2, y = 3x, y = 2x - 1, are all examples of linear functions. The identity function of y = x can also be considered a linear function. Graphically the linear function can be represented by the equation of a line y = mx + c, where m is the slope of the line and c is the y-intercept of the line.

The general form of a linear function in two variables is f(x, y) = ax + by, which is used to represent objective functions in linear programming problems. Here x, and y are variables; and a and b are real numbers.

Quadratic Function

A quadratic function has a second-degree quadratic equation and it has a graph in the form of a curve. The general form of the quadratic function is f(x) = ax² + bx + c, where a ≠ 0 and a, b, c are constant and x is a variable. The domain and range of the quadratic function is ℝ.

The graph of a quadratic equation is a non-linear graph and is parabolic in shape. Examples of quadratic functions are f(x) = 3x² + 5, f(x) = x² - 3x + 2.

Cubic Function

A cubic function has an equation of degree three. The general form of a cubic function is f(x) = ax³ + bx² + cx + d, where a ≠ 0 and a, b, c, and d are real numbers & x is a variable. The domain and range of a cubic function is ℝ.

The graph of a cubic function is more curved than the quadratic function. An example of a cubic function is f(x) = 8x³ + 5x² + 3.

Polynomial Function

The general form of a polynomial function is f(x) = aⁿxⁿ + a^n-1x^n-1 + a^n-2x^n-2+ ..... ax + b. Here n is a nonnegative integer and x is a variable. The domain and range of a polynomial function are R. Based on the highest power (exponent) of the polynomial function, the functions can be classified as a quadratic function, cubic function, etc.

Functions are used in all the topics of maths. The functions have been classified based on the concept in which they have been used. For example, the functions used in trigonometry are called trigonometric functions. There are mainly 4 types of such functions which are explained as follows.

Algebraic Functions

An algebraic function is helpful to define the various operations of algebra. The algebraic function has a variable, coefficient, constant term, and various arithmetic operators such as addition, subtraction, multiplication, and division. An algebraic function is generally of the form of f(x) = aⁿxⁿ + a^{n - 1}x^{n - 1}+ a^n-2x^n-2+ ....... ax + c.

The algebraic function can also be represented graphically. The algebraic function is again classified into the following functions based on their degree:

• Linear functions
• Quadratic functions
• Cubic function
• Polynomial functions

We study about these functions in the next section.

Trigonometric Functions

The six basic trigonometric functions are f(θ) = sin θ, f(θ) = cos θ, f(θ) = tan θ, f(θ) = sec θ, f(θ) = cosec θ. Here the domain value θ is the angle and is in degrees or in radians. These trigonometric functions have been taken based on the ratio of the sides of a right-angle triangle, and are based on the Pythagoras theorem.

Inverse Trigonometric Functions

Further from these trigonometric functions, inverse trigonometric functions have also been derived. The six basic inverse trigonometric functions are f(x) = sin^-1x, f(x) = cos^-1x, f(x) = tan^-1x, f(x) = sec^-1x, f(x) = cosec^-1x, f(x) = cot^-1x. The domain of the inverse trigonometric function contains real number value and its range has angles. The trigonometric functions and the inverse trigonometric functions are also sometimes referred to as periodic functions since the principal values are repeated.

Logarithmic and Exponential Functions

Logarithmic functions have been derived from the exponential functions. The logarithmic functions are considered as the inverse of exponential functions. Logarithmic functions have a 'log' in the function and it has a base. The logarithmic function is of the form y = \(\log_ax \). Here the domain value is the input value of 'x' and is calculated using the Napier logarithmic table. The logarithmic function gives the number of exponential times to which the base has raised to obtain the value of x. The same logarithmic function can be expressed as an exponential function as x = a^y.

The other types of functions apart from the above are as follows:

Modulus Function

The modulus function gives the absolute value of the function, irrespective of the sign of the input domain value. The modulus function is represented as f(x) = |x|. The input value of 'x' can be a positive or a negative expression. The graph of a modulus function lies in the first and the second quadrants since the coordinates of the points on the graph are of the form (x, y), (-x, y).

Rational Function

A function that is composed of two functions and expressed in the form of a fraction is a rational function. A rational fraction is of the form f(x)/g(x), and g(x) ≠ 0. The graphical representation of these rational functions involves horizontal / vertical asymptotes, and the function does not touch the asymptotes.

Signum Function

The signum function helps us to know the sign of the function and does not give the numeric value or any other values for the range. The range of the signum function is limited to {-1, 0, 1}. For the positive value of the domain, the signum function gives an answer of 1, for negative values the signum function gives an answer of -1, and for the 0 value of a domain, the image is 0. The signum function has wide applications in software programming.

Even and Odd Function

The even and odd functions are based on the relationship between the input and the output values of the function. For the negative domain value, if the range is a negative value of the range of the original function, then the function is an odd function. And for the negative domain value, if the range is the same as that of the original function, then the function is an even function.

• If f(-x) = f(x), for all values of x, then the function is an even function. Example: f(x) = x², f(x) = cos x, etc.
• If f(-x) = -f(x), for all values of x, then the function is an odd function. Examples: f(x) = x³, f(x) = sin x, etc.

Periodic Function

The function is considered a periodic function if the same range appears for different domain values and in a sequential manner. The trigonometric functions can be considered periodic functions. For example, the function f(x) = sin x, have a range [-1, 1] for the different domain values of x = nπ + (-1)ⁿx. Similarly, we can write the domain and the range of the trigonometric functions and prove that the range shows up in a periodic manner.

Inverse function

The inverse of a function f(x) is denoted by f^-1(x). For the inverse of a function the domain and range of the given function are changed as the range and domain of the inverse function. The inverse of a function can be prominently seen in algebraic functions and in inverse trigonometric functions. The domain of sin x is R and its range is [-1, 1], and for sin^-1x the domain is [-1, 1] and the range is R. The inverse of a function exists only if it is a bijective function.

If a function f(x) = x², then the inverse of the function is f^-1(x) = \(\sqrt x\).

Greatest Integer Function

The greatest integer function is also known as the step function. The greatest integer function rounds up the number to the nearest integer less than or equal to the given number. Clearly, the input variable x can take on any real value. However, the output will always be an integer. Also, all integers will occur in the output set. Thus, the domain of this function is real numbers ℝ, while its range is integers (ℤ).

The greatest integer function graph is known as the step curve because of the step structure of the curve. For example, when x taking values from [1, 2), the value of f(x) is 1. The greatest integral function is denoted as f(x) = ⌊x⌋.

Composite Function

The composite functions are of the form of gof(x), fog(x), h(g(f(x))), and is made from the individual functions of f(x), g(x), h(x). The composite functions made of two functions have the range of one function forming the domain for another function. Let us consider a composite function fog(x), which is made up of two functions f(x) and g(x).

Here we write fog(x) = f(g(x)). The range of g(x) forms the domain for the function f(x). It can be considered as a sequence of two functions. If f(x) = 2x + 3 and g(x) = x + 1 we have fog(x) = f(g(x)) = f(x + 1) = 2(x + 1) + 3 = 2x + 5.

Every function is associated with a graph that passes vertical line test. The graph of a function depends on its type. For example:

• The graph of a linear function is a line.
• The graph of a quadratic function is 'U' shaped (parabola).
• The graph of sine/cosine function is wavy.
• The graph of an absolute value function is 'V' shaped.

Take a look at the figure below that shows graphs of some other types of functions. Here, all the graphs has horizontal/vertical/both asymptotes.

☛ Related Topics:

• Function Transformations
• Continuous Function
• Graphing Functions Calculator

What are the Types of Functions in Maths?

The types of functions can be broadly classified into four types.

• Based on mapping: One to one Function, many to one function, onto function, one to one and onto function, into function.
• Based on math topics: Algebraic Functions, Trigonometry functions, logarithmic functions.
• Based on degree: Identity function, linear function, quadratic function, cubic function, polynomial function.
• Miscellaneous functions: Modulus function, rational function, signum function, even and odd function, greatest integer function.

How do you Determine the Types of Functions?

The types of functions can be determined based on their mapping, type of math topic, and degree. Further classifying functions into types of functions helps to group and easily understand each of the types of functions. All the trigonometric functions can be grouped under periodic functions. Functions such as the identity function, linear function, quadratic, and cubic function can be grouped under polynomial functions. Thus the function equation y = f(x) is helpful to define the type of function.

What are the Types of Functions Graphs?

The graphs of different types of functions are different. Some of them may have asymptotes too. They might be linear or curvy. All these things depend upon the equation of the function.

• Example 1: The equation of f(x) = x² is a parabola.
• Example 2: The equation of g(x) = |x| is a V-shaped graph.
• Example 3: The equation of h(x) = 1/x has two curves with one vertical asymptote and one horizontal asymptote.

What are the applications of types of functions?

The types of functions have numerous applications in physics, engineering, computer sciences, artificial intelligence, etc. All of these fields aim at connecting one set of data points (domain) to another set of data points(range). Also, the functions help in representing the huge set of data points in a simple mathematical expression of the formal y = f(x).

What Type of Functions is Always Continuous?

The function whose graph can be drawn without lifting a pen is known as a continuous function. The simplest example of a continuous function is the identity function. Other examples of continuous functions are the trigonometric sine function and cosine function.

What are the Various Forms of Representation of Functions?

The functions are generally represented in the form of an equation y = f(x), where the set of x values is the domain and the set of y or f(x) values is the range of the function. The functions can be represented by equations and graphs.

Can any Math Equation be Considered a Function?

No, a relation of the form y = f(x) is considered as a function only if it passes the vertical line test.

What are the other math applications of Types of Functions?

The types of functions have enormous applications in algebra, trigonometry, logarithms, exponents. All the algebraic expressions can be counted as functions as it has an input domain value of x and the output range, which is the answer of the algebraic function. The trigonometric functions also have the angle value as the domain and range value.