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Introduction |
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We consider two sets A and B. We form the Cartesian Product, we form relations. From all the relations, we can select a few which satisfy the rule that each element of the set A is related to only one element of the set B.
When a relation satisfies this rule, it is called a function. |
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Function |
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Any relation on A x B in which (i) no two second elements have a common first element and (ii) every first element has a corresponding second element is called a function. |
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It is denoted by  where y is a function of x. |
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(a) First element is also called
(i) abscissa or
(ii) first component or
(iii) pre-image
(b) Second element is also called
(i) ordinate or
(ii) second component or
(iii) image |
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Representation of a Function |
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A function can be represented by the following methods:
(i) An arrow diagram. (ii) Cartesian graph. (iii) Set-builder notation. |
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Types of Functions (Mapping) |
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The following are the Types of Functions or Mapping:
(1) One-one function
(2) Many-one function
(3) Onto function
(4) Into function. |
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Test for Functions (Summary) |
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The relation should be one-one or many-one, and
Every first element is mapped which means that every pre-image should have an image. |
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More Solved Examples |
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The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its "input") and the other as secondary (the value of the function, or "output"). A function then is a way to associate a unique output for each input of a specified type, for example, a real number or an element of a given set. This definition covers most elementary functions, maps between algebraic structures, such as groups, and between geometric objects, such as manifolds. |
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Summary |
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A function is a relation on A x B is which
(i) no two second elements have a common first element.
(ii) every first element has a corresponding second element.
Every function is either one-one onto or one-one into or many-one onto or many-one into.
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