Function

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A function is a rule that maps one set of values to another set of values. For instance, one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it takes its input value, and squares it to get an output value. Let's call this function $f$ . A common notation to define $f$ is: $f(x) = x^2$ . This tells us that $f$ is a function that squares its argument (its input value). Note that this "rule" can be arbitrarily complicated and doesn't need to be given by a simple formula or description. The only requirement is that $f(x)$ should be uniquely determined by $x$ . The following are examples of functions:

$\displaystyle f(x)=x ^ {2}+2x-2$

$\displaystyle f(x)=\sin(\log{x})$

$\displaystyle f(x)=x^2$ for $\displaystyle x>0$ , otherwise $\displaystyle f(x)= \sin{x}$

$\displaystyle f(x)=p(g(x))$

$\displaystyle g(x)=F'(x)$

Since functions cover such an enormous part of mathematics, we divide this topic into several articles:

History of Functions

Without being used explicitly, the notion of function first appears with the ancient Greeks and Egyptians.

The rigorous definition was stated in the 19th century and is the result of the works of some famous mathematicians: A.L. Cauchy, Leonhard Euler, B. Riemann. With the development of set theory, a new branch of mathematics appeared, mathematical analysis, in which the notion of function has a central role.

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Function

History of Functions

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