Types of Functions: Elements, Equations, Range, and Domain

A function is simply a relationship between an element of one non-empty set with one element of another non-empty set. A relation between a set of inputs and a set of allowable outputs is known as a function, and it has the feature that each input is associated with precisely one output.

• When every element in set A has one end and only one image in set B, then the mapping from set A to set B is a function.
• Let A and B be two non-empty sets. A function can be described as a particular relation that associates each member of set A with a single element of set B.
• There can be no empty set in either A or B.
• For a given input, a function will provide a certain output. 
• So, f: A → B is a function such that (a, b)∈f, and for a∈A there is a special element b∈B.
• Every mathematical statement may be represented as a function if it has an input value and an output result.

Read Also: Class 12 Relation and Function

Key Terms: Domain, Calculus, Particular equation, Function, Into funtion, Onto Function, Polynomial function, Quadratic Functions, Constant Function 

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What is a function? 

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Functions are described as relationships that provide a certain output for a specific input value. A function has a range and a codomain. Typically, f(x) stands for a function with x as its input.

• A function is often expressed as y = f(x). A relationship between sets A and B is called a function.
• Each element in set A has more than one picture in set B and every element in set A has an image in set B.
• Considering two nonempty sets, A and B.
• The following is how to write a function or mapping from A to B.
• According to the rule "A → B," each element a∈A is connected to a certain element b∈B.

Read More: Degree of Polynomial

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Types of Functions - Based on Elements 

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Types of functions based on elements are mentioned as follows:

1. One to One function – A one-to-one function, sometimes called an injective function, is defined as f: A→ B such that each element of set A is related to a unique element of set B. Every element in one domain has a unique image or co-domain element in this case for the provided function.

2. Many to one function - The condition that more than one element of set A is related to the same element of set B may be defined by the function f: A→B. Here, many elements share the same co-domain or picture.

3. Onto functions - Every element in the codomain of an onto function is connected to the domain element. Every element in set B must have a pre-image in set A for a function defined by f: A→B.

4. Into function – Certain components in the co-domain in this case won't have any pre-images. Set B's components are extra and unrelated to any of set A's components. In terms of its qualities, an onto function is completely different from an into function.

5. Constant function - One of the key variations of a many-to-one function is a constant function. Each element in this domain only contains one picture. The form of the constant function is f(x) = K, where K is a real integer.

Read More: Types of Relations

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Types of Functions - Based on Equation 

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Following are the types of functions based on the equation:

1. Identify Function: The function with the same domain and range is the one to identify.
2. Constant Function: The polynomial function of degree zero is a constant function
3. Linear Function: The first-degree polynomial function.
4. Quadratic Function: The degree of two polynomial functions.
5. Cubic Function: The degree of three polynomial functions.

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Types of Functions - Based on Range 

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Types of functions based on the range are:

1. Modulus Function – No matter what the sign of the input domain value is, the modulus function always returns the function's absolute value.f(x) = |x| is the definition of the modulus function.
2. Rational Function – A rational function is any function that consists of two other functions and is stated as a fraction of X. The formula for a rational function is f(x)/g(x), where g(x) ≠ 0.
3. Periodic Function – If the same range emerges for various domain values and sequentially, the function is said to be periodic.
4. Inverse Function – A function's inverse is a function in which the provided function's domain and range are reversed to become those of the inverse function. F^-1(x) stands for the inverse function, f^-1(x).
5. Signum Function – The signum function is a particular kind of function that aids in determining the function's sign but does not provide a numerical value or any other range values.
6. Even and Odd Function – The connection between the input and output values of the function serves as the foundation for the even and odd function types.
7. Composite Function – A composite function is a sort of function that consists of two functions, with one function's domain acting as the range for the other.
8. Greatest Integer Function – The type of function known as the largest integer function rounds a number to the closest integer that is less than or equal to the input value. The expression f(x) = [x] denotes the largest integer function.

Read More: Inverse Function Formula

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Types of Function - Based on Domain 

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Types of functions based on the domain are listed below:

1. Algebraic Function – A function with a limited number of terms including addition, subtraction, multiplication, and division as well as powers and roots of the independent variable x is known as an algebraic equation.
2. Trigonometric Functions – The domain and range of a trigonometric function are comparable to those of other functions. For example sinθ, cosθ, tanθ, cosecθ, secθ, cotθ.
3. Logarithmic Functions – The kind of function that is produced by exponential functions is a logarithmic function. The inverse of exponential functions is thought to be the logarithmic functions.

Read More: Absolute Value Formula

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Things to Remember

• A process or relationship between an element of one non-empty set and an element of another non-empty set is known as a function.
• In other terms, a function is a relation or the linkage of each member 'a' of non-empty set A to at least one element 'b' of another non-empty set B.
• A relationship between a set A (the function's domain) and a different set B (its co-domain), which can be referred to as a function.
• The collection of values that may be used as input into a function while it is defined is referred to as a domain.
• A co-domain is a collection of values that might be the result of a function.
• The range includes all possible values that can be retrieved as the function's output.

• Different mathematical functions can safeguard us against being mistreated, tricked, or exploited in life.