Implicit Function/Examples
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Examples of Implicit Functions
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 $x \in \R$.