Function and its Types

Domain, range, and function expression define the types of functions. The expression used to write a function is the primary defining characteristic. Along with expression, the relationship between the elements of the domain set and the range set determines the type of function. Understanding and learning the different types of functions is easier with the classification of functions.

Almost every mathematical expression which has an input value and a result can be conveniently represented as a function. In this article, we will examine the types of functions, their definitions, and examples.

Different Types of Functions?

Based on factors such as the domain and range of the function, and the expression, y = f(x) is classified into different types of functions. Input is referred to as the domain x value of the functions. Domain values can be numbers, angles, decimals, or fractions. Likewise, the “y” value (which is generally a numeric value) represents the range. There are four types of functions.

1. Based on the Set Elements.
2. Based on Equation.
3. Base on Range.
4. Based on Domain.

On Elements:

1. One One Function
2. Many One Function
3. Onto Function
4. One One and Onto Function
5. Into Function
6. Constant Function

On Equation

1. Identity Function
2. Linear Function
3. Quadratic Function
4. Cubic Function
5. Polynomial Functions

On Range

1. Modulus Function
2. Rational Function
3. Signum Function
4. Even and Odd Functions
5. Periodic Functions
6. Greatest Integer Function
7. Inverse Function
8. Composite Functions

On Domain

1. Algebraic Functions
2. Trigonometric Functions
3. Logarithmic Functions

Greatest Integer Function

The greatest integer function is the step function.In other words, the input variable x can take on any value. The output will always be an integer. Additionally, all integers will appear in the output set. Thus, the domain of this function is real numbers R, while its range is integers (Z).

Algebraic Function

The various algebraic operations define by using algebraic functions. The algebraic function has a variable, coefficient, constant term, and various arithmetic operations 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.

Graphs can also be used to represent algebraic functions. Based on the degree of the algebraic equation, the algebraic function is also referred to as a linear function, quadratic function, cubic function, or polynomial function.

Homogeneous function

Homogeneous functions are functions with multiplicative scaling behavior: if an argument is multiplied by some factor, then its result is also multiplied by some power of this factor. If f: v→W is a function between two vector spaces over a field F, and k is an integer, then * is said to be homogeneous if for all nonzero a belongs to F and v belongs to V. This implies that it is scale-invariant. In vector spaces over real numbers, a slightly more general form of homogeneity is often used, which requires that hold for all α > 0. Homogeneous functions can also be defined for vector spaces with the origin removed, which is used to define sheaves in algebraic geometry. The homogeneous function from S to W can still be defined if S * V is any subset that is invariant under scalar multiplication by elements of the field.

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