The types of functions are determined by the domain, range, and function expression. Functions are primarily defined by the expression they are written in. Furthermore, the relationship between the elements of the domain set and the range set also accounts for the type of function. The classification of functions makes it easier to understand and learn the different types of functions.

It is possible to represent any mathematical expression that has an input value and a result as a function. In this chapter, we will examine types of functions, their definitions, and examples.

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## What are the Different Types of Functions?

A function y = f(x) is classified into different types based on its domain and range, and its expression. Functions have a domain x value, which is their input. A domain value can be a number, angle, decimal, or fraction. Range is defined by the y value or the f(x) value, which is a numeric value. Functions can be divided into the following four types.

- Based on the Set Elements
- Based on Equation
- Based on Range
- Based on Domain

## Representation of Functions

Functions can be represented in three different ways. To showcase the domain values and the range values and their relationship, the functions must be represented. Venn diagrams, graphical formats, and roster forms can be used to represent the functions. Here are the details of each form.

**Venn Diagram: **This is a popular format for representing functions. Venn diagrams are usually shown as two circles connected by arrows. Domain values are presented in one circle and range values in another. Functions define the arrows, and how the arrows connect the different elements in the two circles.

**Graphical Form: **Representing functions graphically with the help of coordinate axes makes them easier to understand. We can understand the changing behavior of the function by representing it graphicaly if it is increasing or decreasing. On the x-axis is plotted the domain or the x value of the function, and on the y-axis is plotted the range or the f(x) value of the function.

**Roster Form:** Domain and range are represented in flower brackets, with the first element of a pair representing the domain, and the second element representing the range. Let’s try to understand this with a simple example. For a function of the form f(x) = x^{2}, the function is represented as {(1, 1), (2, 4), (3, 9), (4, 16)}. The first element is the domain or x value, while the second element is the range or f(x) value of the function.

## Classification of Types of Functions

Various types of functions are further classified to make learning and understanding easier. The types of functions have been further classified into four different types, and are presented as follows.

## Types of Functions – Based on Set Elements

Functions of this type are classified based on the number of relationships between the elements in a domain and those in a codomain. Here are the different types of functions based on set elements.

**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. Injective functions are also called one-to-one functions. Each element of the domain has an associated 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. A many to one function has more than one element with the same co-domain or image. If a many to one function in the codomain is a single value, or if all domain elements are connected to a single element, it is called a constant function.

**Onto Function**

Each codomain element is related to the domain element in an onto function. 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 known as a subjective function.

**One One and Onto Function (Bijection)**

A bijective function is both a one-one function and an onto function. In this case, every element in the domain is connected to a distinct element in the codomain, and every element in the codomain has a pre-image. Also, every element of set A is connected to a distinct element of set B, and none of the elements of set B have been left out.

**Into Function**

The properties of an into function are exactly opposite those of an onto function. There are certain elements in the co-domain that do not have pre-images. Set B consists of excess elements that are not connected to any elements in set A.

**Constant Function**

A constant function is an important type of many-to-one function. In a constant function, all domain elements have the same image. The constant function has 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.

## Types of Function – Based on Equation

Algebraic expressions are also functions, and are derived based on the degree of the polynomial. Functions based on equations are classified into the following equations according to the degree of the variable ‘x’.

- The polynomial function of degree zero is called a Constant Function.
- The polynomial function of degree one is called a Linear Function.
- The polynomial function of degree two is called a Quadratic Function.
- The polynomial function of degree three is a Cubic Function.

Let us understand each of these functions in detail.

**Identity Function**

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

The graph of the identity function is a straight line passing through the origin, equally inclined to the coordinate axes. Since the identity function can have both positive and negative values, it appears in the first and third quadrants of the coordinate axis.

**Linear Function**

A polynomial function with a first-degree equation is a linear function. Linear functions have a straight line graph, and their domain and range are real numbers. 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.

The general form of a linear function is f(x) = ax + b, where a, b are real numbers. Graphically to represent 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.

**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 the the quadratic function is f(x) = ax^{2} + bx + c, where a ≠ 0 and a, b, c are constant & 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) = 3x^{2} + 5, f(x) = x^{2} – 3x + 2.

**Cubic Function**

Cubic functions have an equation of degree three. The general form of a cubic function is f(x) = ax^{3} + bx^{2} + cx +d, where a ≠ 0 and a, b, c, and d are real numbers & x is a variable. R is the domain and range of a cubic function.

The graph of a cubic function is more curved than that of a quadratic function. An example of cubic function is f(x) = 8x^{3} + 5x^{2} + 3.

**Polynomial Function**

The general form of a polynomial function is f(x) = a^{n}x^{n} + a^{n-1}x^{n-1} + a^{n-2}x^{n-2}+ ….. ax + b. The variable x is a nonnegative integer and n is a nonnegative integer. A polynomial function has a domain and range. Depending on the power of the polynomial function, the functions can be classified as a quadratic function, cubic function, etc.

## Types of Functions – Based on Range

According to the range obtained from the given functions, the types of functions have been classified. The types of functions are as follows.

**Modulus Function**

No matter what the sign of the input domain value is, modulus gives the absolute value of the function. The modulus function is represented as f(x) = |x|. The input value of ‘x’ can be a positive or a negative expression. Since the coordinates of the points on the graph are of the form (x, y), (-x, y), the graph of a modulus function lies in the first and second quadrants.

**Rational Function**

Rational functions are composed of two functions and expressed in the form of a fraction. A rational fraction is of the form f(x)/g(x), and g(x) ≠ 0. The functions used in this rational function can be an algebraic function or any other function. Since these rational functions do not touch the axis lines, their graphical representation is similar to the asymptotes.

**Signum Function**

This function tells us only the sign of the function, not the numeric value or any other value for the range. The signum function has a range of {-1, 0, 1}. When a domain has a positive value, the signum function returns a 1, when a domain has a negative value, the signum function returns -1, and when a domain has a zero value, the signum function returns 0. In software programming, the signum function is widely used.

**Even and Odd Function**

Even and odd functions are determined by the relationship between input and output values. If the range is also a negative value of the range of the original function, then the function is odd. If the range of the negative domain value matches the range 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, and if f(-x) = -f(x), for all values of x, then the function is an odd function. An example of even functions are x^{2}, Cosx, Secx, and an example of odd functions are x^{3}, Sinx, Tanx.

**Periodic Function**

If the same range appears for different domain values and sequentially, this is considered a periodic function. The trigonometric functions can also be considered periodic functions. For example, the function f(x) = Sinx, have a range equal to the range of [-1, 1] for the different domain values of x = nπ + (-1)^{n}x. In a similar manner, we can write the domain and range of trigonometric functions and prove that the ranges occur in a periodic manner.

**Inverse function**

The inverse of a function f(x) is denoted by f^{-1}(x). For inverse functions, the domain and range of the given function are reverted to the domain and range of the inverse function. Inverse trigonometric functions and algebraic functions are good examples of inverse functions. The domain of Sinx is R and its range is [-1, 1], and for Sin^{-1}x the domain is [-1, 1] and the range if R. An inverse function exists if it is bijective.

If a function f(x) = x^{2}, then the inverse of the function is f^{-1}(x) = √xx.

**Greatest Integer Function**

The greatest integer function is also referred to as the step function. In the greatest integer function, the number is rounded up to the nearest integer less than or equal to the given number. Input variable x can have any real value. But the output will always be an integer. All integers will also appear in the output set. Hence, the domain of this function will be real numbers R, while its range will be integers (Z).

Due to its step structure, the greatest integer function graph is referred to as the step curve. The greatest integral function is denoted as f(x) = ⌊x⌋. For a function taking values from [1, 2), the value of f(x) is 1.

**Composite Function**

The composite functions are of the form of gof(x), fog(x), h(g(f(x))), and are composed of f(x), g(x), h(x). Composite functions made of two functions have the range of one function forming the domain of the other function. 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.

## Types of Functions – Based on Domain

Mathematical functions are used in all other topics. Based on the types of equations used to define the functions, the functions have been classified. The domain values of these function equations are generally algebraic expressions, trigonometric functions, logarithms, and exponents. Here are three broad types of functions based on domain value.

**Algebraic Function**:

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

The algebraic function can also be represented graphically. Based on the degree of the algebraic equation, the algebraic function can be called a linear function, quadratic function, cubic function, or polynomial function.

**Trigonometric Functions**

Like any other function, trigonometric functions have a domain and range. The six trigonometric functions are f(θ) = sinθ, f(θ) = cosθ, f(θ) = tanθ, f(θ) = secθ, f(θ) = cosecθ. Here the domain value θ is the angle and is in degrees or in radians. These trigonometric functions are based on the Pythagorean theorem and the ratio of the sides of a right-angle triangle.

In addition to these trigonometric functions, inverse trigonometric functions have also been developed. Inverse trigonometric functions have a real number domain and an angle range. Inverse trigonometric functions and trigonometric functions are sometimes referred to as periodic functions since the principal values are repeated.

**Logarithmic Functions**

Logarithmic functions are derived from exponential functions. The inverse of exponential functions is a logarithmic function. There is a ‘log’ in a logarithmic function, and there is a base too. The logarithmic function is of the form y = logaxlogax. In this case, the domain value is the input value of ‘x’ and is calculated using the Napier logarithmic table. The logarithmic function is used to calculate the number of exponential times the base has been raised to obtain the value of x. The same logarithmic function can be expressed as an exponential function as x = a^{y}.

## FAQs

**What are the Various Forms of Representation of Functions?**

Typically, functions are represented by the equation y = f(x), where x is the domain and y or f(x) is the range of the function. Functions with domain and range elements are also represented as Venn diagrams or rosters. In roster form the domain and range of the function are represented as {(x1,f(x1)),(x2,f(x2)),(x3,f(x3))(x1,f(x1)),(x2,f(x2)),(x3,f(x3))}.

**What are the Types of Functions?**

Functions can be broadly classified into four types.

Based on Element: One to one Function, many to one Function, onto function, one to one, and onto function, into function.

Based on Domain: Algebraic Functions, Trigonometry functions, Logarithmic functions.

Based on Range: Modulus function, rational function, signum function, even and odd function, greatest integer function.

Based on Equations: Identical function, linear function, quadratic function, cubic function, polynomial function.

**How do you Determine the Types of Functions?**

Based on the domain, range, and functional equation, one can determine the type of function. In addition, classifying functions into types of functions helps to group and understand each of the types of functions. For instance, all trigonometric functions can be grouped under periodic functions. A polynomial function can be defined as an identity function, a linear function, a quadratic or cubic function. The function equation y = f(x) can be used to identify the type of function.

**What Type of Function is Always Continuous?**

A continuous function is one that has a range for every input value of the domain. A simple example of a continuous function is the identity function, which has an equal range and domain value. Trigonometric sine and cosine functions are also examples of continuous functions.