Types of Functions

The function y = f(x) is divided into many categories based on factors such as the function’s domain and range, as well as the function expression. The functions take input in the form of a domain x value. A number, an angle, a decimal, or a fraction can all be used as domain values. Similarly, the range is the y value or f(x) value (which is usually a numeric value). The types of functions have been divided into the four categories below.

• Based on the Elements of the Set
• Based on the Equation
• Based on the Range 
• Based on the Domain

Different types Based on the elements : 

• One – One functions
• Many one functions
• Onto functions
• Constant functions
• Into functions

Different types Based on Equations :

• Identity functions
• Linear functions
• Quadratic functions
• Cubic functions
• Polynomial functions 

Different types Based on the Domain :

• Algebraic functions
• Trigonometric functions
• Logarithmic functions
• Exponential functions 

Different types Based on Range : 

• Modulus functions
• Rational functions
• Signum functions
• Even and Odd functions
• Periodic functions
• Greatest Integer functions
• Inverse functions 
• Composite functions 

Based on Set Element: 

The number of relationships between the components in the domain and the codomain is used to classify these functions. The following are the various sorts of functions based on set elements.

One – One Functions : 

The function f(x) is a one-to-one function that returns each element of its range from a single unique element from its domain. This means that for each x value, there will be a single y or f value (x).

Onto functions : 

If two or more components in the Domain have the same component in the Range, the function is Onto Function.

Into Functions : 

There is no pre-image in domain X for a function that requires an element from co-domain Y.

Many One Functions : 

If there are two or more separate items in X that have the same image in Y, the function f is said to be a many-one function.

Constant Functions :

A constant function is one of the most common types of many to one functions. A constant function has a single picture for all domain items. The constant function has the form f(x) = K, where K is a positive integer. For a constant function, the same range value of K is produced for different values of the domain(x value).

Based on Equations : 

Identity Functions :

If each element of set A has an image on itself, f (a) =  a ∀ a ∈ A, the function f is termed the identity function.

Polynomial Functions :

A polynomial function is a function in an equation that contains only non-negative integer powers or only positive integer exponents of a variable, such as the quadratic equation or the cubic equation.

Linear Functions :

The degree one polynomial function.

Quadratic Function :

The degree two polynomial function.

Cubic Function :

The degree three polynomial function.

Based on Range : 

Modulus Functions :

Regardless of the sign of the input domain value, the modulus function returns the function’s absolute value. f(x) = |x| is how the modulus function is written. ‘x’ can be a positive or negative expression as an input value. Because the coordinates of the points on the graph are of the form (x, y),(-x, y) the graph of a modulus function is in the first and second quadrants .

SignumFunctions :

The signum function informs us of the function’s sign but does not provide a numeric value or any other range values. The signum function’s range is limited to -1, 0, 1. The signum function returns 1 when the domain value is positive, -1 when the domain value is negative, and 0 when the domain value is zero.

Odd – Even Functions :

The even and odd functions are determined by the relationship between the function’s input and output values. If the range is a negative value of the original function’s range for the negative domain value, the function is an odd function. If the range of the negative domain value is the same as the original function, the function is an even function.

Greatest Integer Functions :

The step function is another name for the biggest integer function. The biggest integer function takes the given number and rounds it up to the next integer that is less than or equal to it. The input variable x can obviously take on any real value. The result, however, will always be an integer. In addition, the output set will contain all integers. As a result, this function’s domain is real numbers R, while its range is integers (Z).

Composite Functions :

The composite functions have the form gof(x), fog(x), h(g(f(x)) and are composed of the individual functions f(x), g(x), h(g(f(x)) (x). The range of one function forms the domain for another function in composite functions made up of two functions.

Rational Functions :

A rational function is a function that is made up of two functions and stated as a fraction. A rational fraction has the form f(x)/g(x), where g(x) is not equal to  zero. An algebraic function or any other function can be employed in this rational function. 

Inverse Functions :

f -1(x) denotes the inverse of a function f(x). The domain and range of the provided function are adjusted to the range and domain of the inverse function when computing the inverse of a function. In algebraic functions and inverse trigonometric functions, the inverse of a function is prominent.

Periodic Function :

If the same range appears for successive domain values in a sequential fashion, the function is considered periodic. Periodic functions can be considered trigonometric functions.

Based on Domain : 

Algebraic Functions :

An algebraic function is a type of function that can be used to define various algebraic operations. A variable, coefficient, constant term, and different arithmetic operators like addition, subtraction, multiplication, and division are all included in this function.

Trigonometric Functions :

The trigonometric function has a domain and range that are similar to those of any other function. The following are the six trigonometric functions:

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

Logarithmic Functions :

The sort of function that is derived from exponential functions is called a logarithmic function. Exponential functions are considered the inverse of logarithmic functions.

Conclusion :

In mathematics, a function is an expression, rule, or law that describes the relationship between one variable (the independent variable) and another variable (the dependent variable) (the dependent variable). In mathematics, functions are everywhere, and they’re crucial for articulating physical links in the sciences.