Implicitly defined function

An implicitly defined function is a function that is presented as the solution of some equation or system of equations, rather than being given by an explicit formula.

Equations defining functions implicitly can sometimes be solved to give the function explicitly. However, this is not generally true, and it can be difficult or impossible to express simple implicitly defined functions in explicit form.

Examples

The equation $y + \tan^{-1} y = x$ implicitly defines $y$ as a function of $x$ . This equation cannot be solved for $y$ as an elementary function in terms of $x$ .

The equation $y^3 + 2y^2 + 2y + x^2 + x = 0$ implicitly defines $y$ as a function of $x$ over the real numbers. This equation can be solved for $y$ in terms of $x$ by using the cubic formula, but the resulting expression is very unpleasant to work with.

The equation $w \cdot e^w = x$ for $x \geq -\frac{1}{e}$ and $w \geq -1$ implicitly defines $w$ as a function of $x$ . This function is known as the Lambert W function. This equation cannot be solved for $w$ as an elementary function in $x$ , but this function is of sufficient importance that it has been given its own name.

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Implicitly defined function

Examples

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