Implicit Function/Examples

Examples of Implicit Functions

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

Consider the equation:

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

where:

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

$r \in \R_{>0}$ is a strictly positive real constant.

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

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 $x \in \R$.