IMPLICIT FUNCTION Meaning and
Definition
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An implicit function is a mathematical concept that defines a relationship between two or more variables by means of an equation where the dependent and independent variables are not explicitly separated. In other words, instead of expressing one variable directly in terms of the others, an implicit function describes the relationship between variables through an equation in which the dependent variable is equated to a function of the independent variables.
The characteristic feature of an implicit function is that it does not explicitly solve for one variable in terms of the others, like in an explicit function. Instead, it provides an equation where the variables are interrelated without isolating any particular variable. This often gives rise to equations that cannot be easily solved algebraically.
Implicit functions are commonly encountered in various branches of mathematics, including calculus, differential equations, and geometry. They allow for the representation of complex relationships between variables that cannot be adequately described by an explicit function. Implicit functions are particularly useful in cases involving curves, surfaces, or multidimensional spaces where the explicit expression of a function may not exist or be easily obtainable.
Solving implicit functions typically involves techniques such as implicit differentiation, which allows for finding derivatives without explicitly solving for the dependent variable. Numerical methods and computer algorithms may also be employed to approximate solutions or plot the graphs of implicit functions.
Etymology of IMPLICIT FUNCTION
The word "implicit" originates from the Latin word "implicitus", which means "related, involved, entangled, or folded in". It is derived from the verb "implicare", which means "to involve, entwine, or entangle". The term "function" has its roots in the Latin word "functio", which means "performance, execution, or discharge". The word "implicit function" combines these two terms, referring to a mathematical function that is expressed indirectly or without a specific definition.
