Types of Functions

The range, domain, and function expression – all have a role in determining the types of functions. The primary defining criterion for a function is the expression used to write it. The kind of function is determined by the relationship between the members of the range set and the items of domain set and expression. The classification of functions aids in the understanding and learning of various functions. A function can represent any mathematical expression with input and output values. The function y=f(x) is divided into multiple categories based on factors such as the function’s domain and range and the function expression.

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 four categories based on:

• Set Elements
• Equation
• Domain
• Range

Function Types Based on the Set Elements

One-to-One Function

A one-to-one function is defined as f:AB, which connects each element in set A to a unique element in set B. 

Many-to-One Function

The function f:AB defines a many-to-one function in which more than one element of set A is associated with the same element in set B. More than one element has the same co-domain or image in a many-to-one function. 

Onto Function

Every co-domain element is associated with the domain element in an onto function. Every element in set B has a pre-image in set A for a function defined by f:AB. 

One-to-One and Onto Function

If a function f is both One-to-One and Onto, it is called One-to-One and Onto or Bijective.

Into Function

An Into function is the polar opposite of the Onto function. In the co-domain, some elements do not have a pre-image.

Constant Function

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.

Function Types Based on the Equations

Identity Function

Assume R is a set of real numbers. The function f:RR is known as the Identity function if it is defined as f(x)=x or y=x, for xR. R is the domain, and R is the range.

Linear Function

A first-degree equation polynomial function is referred to as a linear function. A linear function’s domain and range are both real numbers and have a straight line graph. 

Quadratic Function

A quadratic function has a graph in the form of a curve and a second-degree quadratic equation. 

Cubic Function

A cubic function has a three-degree equation. A cubic function’s domain and range are R. 

Polynomial Function

A polynomial function has a general form:f(x)=anxn+an-1xn-1+an-2xn-2+………ax+b. x is a variable, and n is a nonnegative number.

Function Types Based on the Domain

Algebraic Function

An algebraic function is useful for defining the various algebraic operations. 

Trigonometric Function

Trigonometric functions, like any other function, have a domain and range. f()=sin, f()=cos, f()=tan, f()=sec, f()=cot, and f()=cosec are the six trigonometric functions. The angle is the domain value , which can be expressed in degrees or radians.

Logarithmic Function

Exponential functions were used to create Logarithmic functions. The inverse of exponential functions is called logarithmic functions. 

Function Types Based on the Range

Modulus Function

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.

Rational Function

A rational function is a function that is made up of two functions and is stated as a fraction. An algebraic function or any other function can be employed in this rational function.

Signum Function

A signum function informs us of the function’s sign but does not provide a numeric value or any other range values.

Even and Odd Function

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. 

Periodic Function

If the same range appears for successive domain values in a sequence, the function is considered periodic.

Inverse Function

When computing the inverse of a function, the range and domain of the supplied function are modified to the range and domain of the inverse function.

Greatest Integer Function

Step function is another name for the Greatest Integer function. The greatest integer function takes the given number and rounds it up to the next integer that is less than or equal to it.

Composite Function

The range of one function forms the domain for another function in Composite functions consisting of two functions.

Conclusion

Functions are one of the integral parts of various fields related to mathematics as it makes calculation easy and simple to understand. Different types of functions can be used in different situations due to their properties. Functions are used by engineers and scientists to solve real-life problems and implement them in their projects.