In mathematics, an implicit equation is a relation of the form R(x1,…,xn)=0,{\displaystyle R(x_{1},\dots ,x_{n})=0,}R(x1,…,xn)=0, where RRR is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x2+y2−1=0.{\displaystyle x^{2}+y^{2}-1=0.}x2+y2−1=0.

An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. For example, the equation x2+y2−1=0{\displaystyle x^{2}+y^{2}-1=0}x2+y2−1=0 of the unit circle defines yyy as an implicit function of xxx if −1≤x≤1−1 ≤ x ≤ 1−1≤x≤1, and one restricts yyy to nonnegative values.

The implicit function theorem provides conditions under which some kinds of relations define an implicit function, namely relations defined as the indicator function of the zero set of some continuously differentiable multivariate function.

1 Examples
- 1.1 Inverse functions
- 1.2 Algebraic functions
2 Caveats
3 Implicit differentiation
- 3.1 Examples
- - 3.1.1 Example 1
  - 3.1.2 Example 2
  - 3.1.3 Example 3
- 3.2 General formula for derivative of implicit function
4 Implicit function theorem
5 In algebraic geometry
6 In differential equations
7 Applications in economics
- 7.1 Marginal rate of substitution
- 7.2 Marginal rate of technical substitution
- 7.3 Optimization
8 See also
9 References
10 Further reading
11 External links

1 Examples

1.1 Inverse functions

A common type of implicit function is an inverse function. Not all functions have a unique inverse function. If ggg is a function of xxx that has a unique inverse, then the inverse function of ggg, called g−1g^{−1}g−1, is the unique function giving a solution of the equation
y=g(x){\displaystyle y=g(x)}y=g(x)
for xxx in terms of yyy. This solution can then be written as
x=g−1(y).{\displaystyle x=g^{-1}(y)\,.}x=g−1(y).
Defining g−1g^{−1}g−1 as the inverse of ggg is an implicit definition. For some functions ggg, g−1(y)g^{−1}(y)g−1(y) can be written out explicitly as a closed-form expression — for instance, if g(x)=2x−1g(x) = 2x − 1g(x)=2x−1, then g−1(y)=1/2(y+1)g^{−1}(y) = 1/2(y + 1)g−1(y)=1/2(y+1). However, this is often not possible, or only by introducing a new notation (as in the product log example below).

Intuitively, an inverse function is obtained from ggg by interchanging the roles of the dependent and independent variables.

Example: The product log is an implicit function giving the solution for xxx of the equation y−xex=0.y − xe^x = 0.y−xex=0.

1.2 Algebraic functions

Main article: Algebraic function

An algebraic function is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable xxx gives a solution for yyy of an equation
an(x)yn+an−1(x)yn−1+⋯+a0(x)=0,{\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0\,,}an(x)yn+an−1(x)yn−1+⋯+a0(x)=0,
where the coefficients ai(x)a_i(x)ai(x) are polynomial functions of xxx. This algebraic function can be written as the right side of the solution equation y=f(x)y = f(x)y=f(x). Written like this, fff is a multi-valued implicit function.

Algebraic functions play an important role in mathematical analysis and algebraic geometry. A simple example of an algebraic function is given by the left side of the unit circle equation:
x2+y2−1=0.{\displaystyle x^{2}+y^{2}-1=0\,.}x2+y2−1=0.
Solving for yyy gives an explicit solution:
y=±1−x2.{\displaystyle y=\pm {\sqrt {1-x^{2}}}\,.}y=±1−x2.
But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as y=f(x)y = f(x)y=f(x), where fff is the multi-valued implicit function.

While explicit solutions can be found for equations that are quadratic, cubic, and quartic in yyy, the same is not in general true for quintic and higher degree equations, such as
y5+2y4−7y3+3y2−6y−x=0.{\displaystyle y^{5}+2y^{4}-7y^{3}+3y^{2}-6y-x=0\,.}y5+2y4−7y3+3y2−6y−x=0.
Nevertheless, one can still refer to the implicit solution y=f(x)y = f(x)y=f(x) involving the multi-valued implicit function fff.

2 Caveats

Not every equation R(x,y)=0R(x, y) = 0R(x,y)=0 implies a graph of a single-valued function, the circle equation being one prominent example. Another example is an implicit function given by x−C(y)=0x − C(y) = 0x−C(y)=0 where CCC is a cubic polynomial having a “hump” in its graph. Thus, for an implicit function to be a true (single-valued) function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a true function only after “zooming in” on some part of the xxx-axis and “cutting away” some unwanted function branches. Then an equation expressing yyy as an implicit function of the other variables can be written.

The defining equation R(x,y)=0R(x, y) = 0R(x,y)=0 can also have other pathologies. For example, the equation x=0x = 0x=0 does not imply a function f(x)f(x)f(x) giving solutions for yyy at all; it is a vertical line. In order to avoid a problem like this, various constraints are frequently imposed on the allowable sorts of equations or on the domain. The implicit function theorem provides a uniform way of handling these sorts of pathologies.

3 Implicit differentiation

3.1 Examples

3.1.1 Example 1

3.1.2 Example 2

3.1.3 Example 3

3.2 General formula for derivative of implicit function

4 Implicit function theorem

5 In algebraic geometry

6 In differential equations

7 Applications in economics

7.1 Marginal rate of substitution

7.2 Marginal rate of technical substitution

7.3 Optimization

8 See also

9 References

10 Further reading

11 External links

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Implicit function

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