In mathematics, the function can be defined as an expression or rule that defines the relationship between the variables. There is mainly two kinds of variables that is dependent and independent one. Function is defined to be ubiquitous in mathematics and is very much important for the formulation of relationship in science. In the year 1837, the latest definition of function was given by Peter Dirichlet. The given relationship is commonly symbolized as y=f(x). Also, another abbreviated symbol that is g(x) and P(x) is mainly used for representing the function of independent of variable x.  It is mainly done in some specific condition which is completely unknown or unspecified. In other words, a function can be defined as a relationship where the quantity that is dependent variable completely depends or is determined by the help of another quantity that is an independent variable. It can be said that the dependent variable as a function of an independent variable.

Some of the commonly used mathematical formulas are defined as the known function. The formula for the circle of the area is given by A = πr². In this, dependent variable A is considered to be a function of an independent variable which is of various r. The function requires involvement of two or more variables which are considered to be common in mathematics. In the formula of area of a triangle, A= ½ (based * Height). In the given example, a physical constraint that is force is the independent variable which is found to be a positive number. There are many instances when the independent variable is allowed to have negative values which are any real number. Then it is known to be a real-valued function.

The formula of area of a circle is given by a polynomial function. The general form of the given function is

P(x) = a0 + a₁x + a₂x²+⋯+ anxn,

Where all the co-efficient that is (a₀, a₁, a₂,…, an) is provided. All the counting power of x are nothing but the counting number (1,2,3….). Polynomial function has completely studied in the beginning times because of versatility which requires the involvement of real numbers. It can be closely approximated by the help of polynomial function. There are many polynomial functions is completely characterized by the highest for an independent variable. Some of special names that are known common are the powers of one to five that is linear to quantic.

A polynomial function can be provided by a geometrical representation which is done by the help of analytical geometry. The independent variable X is plotted along the X-axis (horizontal line) and the dependent variable y is put on the y-axis (vertical line). Graph function comes up with two co-ordinates (x, y) where y = f(x). Another kind of function which has been studied right from the antiquity are a trigonometric function that is Sin X and Cos X, which is defined as the measure of an angle. As a result of periodic nature, trigonometric functions are mainly used like a model which repeats or cycles. Trigonometric functions are a non-algebraic function which is also known as a transcendental function.

The only practical application of the given function is the variable which is defined as a complex number which is not that much easy to explain. It is nevertheless very much extensive. It mainly used in various domains like electrical engineering and aerodynamics. A complex variable is mainly represented in the form z=X+iY where i is defined as an imaginary number. The value of i is equal to the square root of -1 which have two variables x and y considered as real variables. The possible method of splitting the complex function is in between two parts that are real and imaginary.

An individual can have an inverse function by interchanging the roles of independent and dependent variables for a given function. Inverse function performs as per the name where they need to undo action of function so that they ca return variable to original state. For a given f(x) there exist a function g(y) in such a way g(f(x)) = x and f(g(y))=y then g is defined as the inverse function of the given function f and denoted by f^-1 . Hereby convention the variables are completely interchanged. For given example, the function f(x)= 2x comes up with inverse function f^-1(x) = x/2.

A given function f: A → B can be stated as one to one function if f (a₁) = f (a₂ )

Where (a₁ )= (a₂ ), (a₁ ) (a₂ ) ∈ A

Let us consider A = {(a 1 )( a 2 )( a 3 )( a 4 )} and B = {( b 1 )( b 2 )( b 3 )( b 4 )}

Fig 1: One to One Function

For example: The given function provided below can be taken to be one to one.

1. Each state of India assigns its capital city.

Ans: This is not a one to one function as each state of India comes up with different capital.

2. Function f = {(2,4), (3,6), (-1, -7)}

Ans: This function is one to one just because of one to one as each of the value range comes up with a different value of domain.

3. F(x) = |x|

Ans: In this function, we need to check whether the given function is one to one or not. Now, some of the core values of x (domain) and given function we need to find the value of range.

From the above table the point can be noted that the value in the given range repeats so it is not considered to be one to one function.

Fig 2: Three Graphs for Determining One to One function

In the 1^st graph, if horizontal line is drawn then the line cuts the graph at the given point only one point. So the 1^st graph can be stated as one to one graph function.

In the 2^nd graph, if a horizontal line is drawn then the line cut the graph at any two point so second graph at point is 1 to 1 graph function.

In the 3^rd graph,  if a horizontal line is drawn then the given line cuts the graph into two given point so it non a one to one function.