The term ‘function’ was originated in year 1673 by Gottfried Lebiniz. Functions are the equations that relate an output with an input. The value of the output depends on the value of input. A function is of the form y = f(x) where for every value of x, there is only one value of y. For y = f(x), x is the independent variable whose value is not dependent on any other value and ‘y’ is a dependent variable whose value depends on value of ‘x’. For example: - if we have f(x) = x + 1, we can choose any random value of x and for every value of x, there is a different value of ‘y’.
The input and the output of a function make an ordered pair and is written as like this: -
(Input, Output)
Therefore a function is sometimes called a set of ordered pairs. One input produces only one output of the function. Output is always dependent on the value of the input which is always independent.
Types of functions
There are many types of functions, the important ones are listed below: -
1. Explicit functions: A function is said to be an explicit function if the dependent variable is fully expressed in terms of independent variable. For example: - f(x) = x^2 + 3x – 5 is an explicit function because their relation is such that value of f(x) depends on value of ‘x’.
2. Implicit functions: A function is said to be an implicit function if the dependent variable is not expressed in terms of independent variable. For example: - 5x^2 + 5y^2 = 25 is an implicit function because their relation is not in that term in which we can say that y is dependent on ‘x’.
3. Injective functions: A function is said to be injective when every element of its domain is mapped to at most one element of its co domain. These are also called one to one function as one element is linked to only one other element and one to many is not possible. That means for every value of ‘f’ there is only one value of ‘x’ that corresponds to ‘f’.
For example: - f(x) = x + 8 is one to one as for every possible value of f(x) there is one value of x.
4. Bijective functions: A function is said to be bijective when every element of its co domain is mapped to one elemnt of its domain. These are also called onto functions and one to one functions.
The input and the output of a function make an ordered pair and is written as like this: -
(Input, Output)
Therefore a function is sometimes called a set of ordered pairs. One input produces only one output of the function. Output is always dependent on the value of the input which is always independent.
Types of functions
There are many types of functions, the important ones are listed below: -
1. Explicit functions: A function is said to be an explicit function if the dependent variable is fully expressed in terms of independent variable. For example: - f(x) = x^2 + 3x – 5 is an explicit function because their relation is such that value of f(x) depends on value of ‘x’.
2. Implicit functions: A function is said to be an implicit function if the dependent variable is not expressed in terms of independent variable. For example: - 5x^2 + 5y^2 = 25 is an implicit function because their relation is not in that term in which we can say that y is dependent on ‘x’.
3. Injective functions: A function is said to be injective when every element of its domain is mapped to at most one element of its co domain. These are also called one to one function as one element is linked to only one other element and one to many is not possible. That means for every value of ‘f’ there is only one value of ‘x’ that corresponds to ‘f’.
For example: - f(x) = x + 8 is one to one as for every possible value of f(x) there is one value of x.
4. Bijective functions: A function is said to be bijective when every element of its co domain is mapped to one elemnt of its domain. These are also called onto functions and one to one functions.
