Definition:Implicit Function

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Definition

Consider a (real) function of two independent variables $z = \map f {x, y}$.

Let a relation between $x$ and $y$ be expressed in the form $\map f {x, y} = 0$ defined on some interval $\mathbb I$.


If there exists a function:

$y = \map g x$

defined on $\mathbb I$ such that:

$\forall x \in \mathbb I: \map f {x, \map g x} = 0$

then the relation $\map f {x, y} = 0$ defines $y$ as an implicit function of $x$.


General Definition

Let:

$f: \R^{n + 1} \to \R, \tuple {x_1, x_2, \ldots, x_n, z} \mapsto \map f {x_1, x_2, \ldots, x_n, z}$

be a real-valued function on $\R^{n + 1}$, where:

$\tuple {x_1, x_2, \ldots, x_n} \in \R^n, z \in \R$

Let a relation between $x_1, x_2, \ldots, x_n$ and $z$ be expressed in the form:

$\map f {x_1, x_2, \ldots, x_n, z} = 0$

defined on some subset $S \subseteq \R^n$.


If there exists a function $g: S \to \R$ such that:

$\forall \tuple {x_1, x_2, \ldots, x_n} \in S: z = \map g {x_1, x_2, \ldots, x_n} \iff \map f {x_1, x_2, \ldots, x_n, z} = 0$

then the relation $\map f {x_1, x_2, \ldots, x_n, z} = 0$ defines $z$ as an implicitly defined function of $x_1, x_2, \ldots, x_n$.


Examples

Example: $x^2 + y^2 - 25 = 0$

Consider the equation:

$(1): \quad x^2 + y^2 - 25 = 0$

where $x, y \in \R$ are real variables.


Then $(1)$ defines $y$ as an implicit function of $x$ on the closed interval $\closedint {-5} 5$.


Example: $x^2 + y^2 + 1 = 0$

Consider the equation:

$(1): \quad x^2 + y^2 + 1 = 0$

where $x, y \in \R$ are real variables.


Solving for $y$, we obtain:

$y = \pm \sqrt {-1 - x^2}$

and it is seen that no $y \in \R$ can satisfy this equation.


Hence $(1)$ does not define a real function.


Example: $x^3 + y^3 - 3 x y = 0$

Consider the equation:

$(1): \quad x^3 + y^3 - 3 x y = 0$

where $x, y \in \R$ are real variables.


Then $(1)$ defines $y$ as an implicit function of $x$ for all $x \in \R$.


Also see

For sufficient conditions for the existence of such functions:

  • Results about implicit functions can be found here.


Sources