Function and types of functions

Function and Types of Functions

What is a Function?

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. A function f from a set X to a set Y is denoted by f: X -> Y, where X is called the domain and Y is the codomain.

A function f associates each element x in the domain X to a single element y in the codomain Y. This is often written as y = f(x).

Types of Functions

Functions can be classified in several ways, based on their properties. Here are some of the most common types of functions:

1. Injective (One-to-One) Functions

An injective function, also known as a one-to-one function, is a function where each element of the codomain is mapped to by at most one element of the domain. In other words, no two different elements in the domain have the same image in the codomain.

Mathematical Definition:

A function f: X -> Y is injective if for every a and b in X, if f(a) = f(b), then a = b.

2. Surjective (Onto) Functions

A surjective function, also known as an onto function, is a function where every element of the codomain is the image of at least one element of the domain.

Mathematical Definition:

A function f: X -> Y is surjective if for every y in Y, there exists at least one x in X such that f(x) = y.

3. Bijective (One-to-One and Onto) Functions

A bijective function is both injective and surjective. This means that every element of the domain maps to a unique element of the codomain, and every element of the codomain is mapped to by exactly one element of the domain.

Mathematical Definition:

A function f: X -> Y is bijective if it is both injective and surjective.

4. Constant Functions

A constant function is a function where the output value is the same for every input value in the domain.

Mathematical Definition:

A function f: X -> Y is constant if there exists a c in Y such that for every x in X, f(x) = c.

5. Identity Functions

The identity function is a function that always returns the same value that was used as its argument.

Mathematical Definition:

The identity function id: X -> X is defined by id(x) = x for all x in X.

6. Polynomial Functions

Polynomial functions are functions that can be expressed in the form of a polynomial.

Mathematical Definition:

A polynomial function f: R -> R is given by f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where a_i are constants.

7. Rational Functions

Rational functions are ratios of polynomial functions.

Mathematical Definition:

A rational function f: X -> Y is given by f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions and q(x) is not zero.

8. Exponential and Logarithmic Functions

Exponential functions involve an exponent that is a variable, while logarithmic functions are the inverses of exponential functions.

Mathematical Definitions:

  • Exponential function: f(x) = a^x, where a is a positive constant.
  • Logarithmic function: f(x) = log_a(x), where a is the base of the logarithm.

9. Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides.

Common Trigonometric Functions:

  • Sine function: f(x) = sin(x)
  • Cosine function: f(x) = cos(x)
  • Tangent function: f(x) = tan(x)

10. Piecewise Functions

Piecewise functions are defined by different expressions for different intervals of the domain.

Example:

f(x) = {
  x^2, if x < 0
  x + 2, if x >= 0
}

Comparison Table

Property/Type Injective Surjective Bijective Constant Polynomial
Definition One-to-one mapping Onto mapping One-to-one and onto Same output for all inputs Expressed as a polynomial
Example f(x) = 2x (for x in R) f(x) = x^2 (for x in R+) f(x) = x + 1 (for x in R) f(x) = 7 (for x in R) f(x) = x^2 + 3x + 2
Every input has unique output? Yes No Yes Yes No
Covers entire codomain? No Yes Yes No No
Inverse exists? Yes, but not necessarily onto No, but can be restricted Yes No Depends on degree

Examples

  1. Injective Function:

    • f(x) = 2x for x in R
    • For every y in the codomain, there is at most one x such that f(x) = y.

  2. Surjective Function:

    • f(x) = x^2 for x in R+
    • Every non-negative real number is the square of some non-negative real number.

  3. Bijective Function:

    • f(x) = x + 1 for x in R
    • Every real number y can be written as y = x + 1 for a unique real number x.

  4. Constant Function:

    • f(x) = 4 for x in R
    • No matter what value of x is chosen, the output is always 4.

  5. Polynomial Function:

    • f(x) = x^3 - x + 6
    • This is a cubic polynomial function.

Understanding the different types of functions and their properties is crucial for solving various mathematical problems and is a fundamental concept in calculus, algebra, and other areas of mathematics.