Implicit Functions
An implicit function is a relationship between variables defined by an equation of the form F(x, y) = 0, where neither variable is isolated on one side. Unlike explicit functions such as y = 2x + 3, where y is written directly in terms of x, an implicit relation entangles the variables together in a single equation. The circle equation x² + y² = 25 is a classic example: you cannot write y purely in terms of x with a single formula without introducing a ± sign, yet the equation still perfectly describes a well-defined curve in the plane. Implicit functions appear throughout mathematics and its applications whenever a constraint links two or more quantities without a clean algebraic separation. In linear algebra, linear implicit relations take the form ax + by = c, which defines a line without favouring either variable as the 'input.' Understanding implicit functions builds the foundation for topics ranging from calculus (implicit differentiation) to multivariable analysis and the Implicit Function Theorem, one of the deepest results in advanced mathematics. For high school students, the key skill is recognising that an equation in x and y implicitly defines y as a function of x (at least locally near most points), being able to find specific values, and understanding when a relation fails to be a true function. This article focuses on the linear algebra setting, where implicit relations are linear equations, and lays groundwork for broader study.
- Explicit vs Implicit Form
- Linear Implicit Relations
- Checking Whether a Point Satisfies an Implicit Relation
- Finding y Given x (and Vice Versa)
- When Implicit Relations Fail to Be Functions
- The Implicit Function Theorem (Preview)
Explicit vs Implicit Form
An explicit function writes the dependent variable alone: y = f(x). An implicit relation writes an equation involving both variables: F(x, y) = 0. For example, 3x + 2y − 6 = 0 is the implicit form of the line y = −(3/2)x + 3. Every explicit function y = f(x) can be written implicitly as F(x,y) = y − f(x) = 0, but not every implicit equation can be solved explicitly with a single neat formula.
Linear Implicit Relations
The general linear implicit relation in two variables is ax + by = c, where a, b, c are constants and not both a and b are zero. If b ≠ 0, we can isolate y to get y = (c − ax)/b, recovering an explicit form with slope −a/b and y-intercept c/b. If b = 0, the equation ax = c defines a vertical line x = c/a, which is NOT a function (it fails the vertical line test), illustrating that implicit relations can describe things beyond ordinary functions.
Checking Whether a Point Satisfies an Implicit Relation
To test if a point (x₀, y₀) lies on the curve F(x, y) = 0, substitute directly: compute F(x₀, y₀) and check whether it equals zero. For example, does (2, 1) lie on 3x + 2y − 8 = 0? Substitute: 3(2) + 2(1) − 8 = 6 + 2 − 8 = 0. Yes! The point satisfies the equation and lies on the line.
Finding y Given x (and Vice Versa)
Even without converting to explicit form, you can find specific values. Given 5x − 3y = 15 and x = 6: substitute to get 5(6) − 3y = 15, so 30 − 3y = 15, giving 3y = 15 and y = 5. The point (6, 5) is on the line. This shows that an implicit equation still lets you compute values — it just requires solving an equation each time rather than simply evaluating a formula.
When Implicit Relations Fail to Be Functions
A relation defines y as a function of x only if each x value corresponds to exactly one y value. The vertical line x = c produces infinitely many y values for a single x — not a function. Non-linear implicit relations like x² + y² = r² give two y values for most x values (one positive, one negative), so the full circle is not a function, though the top semicircle (y = √(r²−x²)) and bottom semicircle are each functions individually. Restricting the domain to get a true function is called 'extracting a branch.'
The Implicit Function Theorem (Preview)
The Implicit Function Theorem guarantees that near a point (x₀, y₀) satisfying F(x, y) = 0, as long as ∂F/∂y ≠ 0 at that point, there exists a local explicit function y = g(x) that describes the curve. For a linear relation ax + by = c, this condition is simply b ≠ 0 (the coefficient of y). This theorem justifies why we can 'pretend' y is a function of x and differentiate implicitly, even when we cannot solve for y globally.
The Key Equation
- a — Coefficient of x (dimensionless): The scalar multiplying the variable x; related to the slope of the line when b ≠ 0.
- b — Coefficient of y (dimensionless): The scalar multiplying the variable y; if b = 0, the equation defines a vertical line and y is not determined.
- c — Constant term (dimensionless): The right-hand side constant that shifts the line away from the origin.
- F(x,y) — Implicit function (dimensionless): The two-variable function whose zero set, F(x,y) = 0, defines the curve or line.
Worked Example
Problem. The implicit relation 4x − 3y = 12 defines a line. (a) Rewrite it in explicit form. (b) Find the x-intercept and y-intercept. (c) Verify that the point (3, 0) lies on the line. (d) Find y when x = −3.
Answer. Explicit form: y = (4/3)x − 4; x-intercept (3, 0); y-intercept (0, −4); point (3,0) verified; when x = −3, y = −8.
Origins · 1 minRené Descartes and how this was discovered
Descartes invented the Cartesian coordinate system, making it possible to represent geometric curves — including those defined by implicit equations like x² + y² = r² — as algebraic equations in two variables. His 1637 appendix 'La Géométrie' unified algebra and geometry, laying the foundation for analytic geometry and the modern study of implicit relations and functions. Descartes famously spent his mornings lying in bed thinking — a habit he said was essential to his philosophy. He reportedly had a secret daughter named Francine, whose early death at age five devastated him.
Related Concepts
- Linear Equations in Two Variables — Linear implicit relations of the form ax + by = c are precisely linear equations in two variables; mastering their forms is the direct prerequisite for understanding implicit functions.
- Slope-Intercept Form — Converting an implicit linear relation to explicit slope-intercept form y = mx + b is the fundamental algebraic skill linking both representations of a line.
- Implicit Differentiation — Implicit differentiation extends the study of implicit functions into calculus, allowing us to find the slope of curves defined by F(x, y) = 0 without converting to explicit form.
- Systems of Linear Equations — A system of two implicit linear relations ax + by = c and dx + ey = f asks for points satisfying both equations simultaneously, directly applying the concept of implicit relations.