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The first real approximation of the modern concept of function has been attributed by many mathematical historians to Nicole Oresme, who described the laws of nature as a relation between two magnitudes.
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he began to approach the modern definition of function by studying movement and trying to understand the relationship within its variables.
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René Descartes began to develop analytical geometry, basing his ideas on the relationship between the magnitudes that had been studied many centuries before.
He defined it as an analytical expression composed in any way by a variable quantity and constant variables of numbers or quantities -
Johann Bernoulli defined it as a quantity formed in some way by indeterminate and constant quantities.
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Leonhard Euler published the book Introductio in analysin infinitorum and the term function got famous.
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Euler change the definition
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The french mathematician Edouard Goursat came up with the most popular definition until this day.