Implicit Function/Examples/x^2 + y^2 + 1 = 0

Examples of Implicit Functions

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.

Sources

• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text D$: Implicit Function