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Properties of Functions

A function is that type of relation in which a domain element corresponding to one element exactly in the range set. However, more than one element in domain set can correspond to a single and same element in the range; still the relation is a function.
 
This can be more easily understood by studying the graphs of the relations.
 

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Basic properties of Functions

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The first two parameters that are connected with any function are the domain and range. The domain of a function means the set or sets of real values of the variable for which the function is defined. The range is the set or sets of values of the function from a minimum level to maximum level in the domain of the function. Usually these two are expressed in interval notation.

A square bracket is used if the value at that point is included and a round bracket is used when it is not be included. The interval of all real numbers is represented as (-∞,∞).

A function may not be defined before or after certain value of the variable.
For example, the function f(x) = √(1 + x) is not defined for all values of ‘x’ before -1 and the function f(x) = √(1 - x) is not defined for all values of ‘x’ after 1. But in many cases a function may not be defined only for a particular or for some values of the variable.

At such places the function is said to be have discontinuities.

One to One and Onto Functions

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At the beginning we said that a relation can be called a function if one domain element corresponds to only element in the range and further extended that even more than one domain element can correspond to a single range element. In the former case the function is called as 'One to one' and the later case is referred as 'many to one'. For example, f(x) = 2x + 1 is an one to one function. But, f(x) = x2 is a many to one function', because for both x = -2 and x = 2, f(x) is same as 4.

If all the elements of range sets are used by all the elements of domain set (even if it is many to one), the function is called onto. Again, f(x) = 2x + 1 is onto because the range is open as (-∞, ∞). But in case of f(x) = x2, the range is not covering the negative values. Hence this function is not onto.

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