Implicit functions

In some situations we do not get a function explicitly, but by an equation that features both the unknown and the function. For instance, the equation x + y = 1 defines the function given by y = 1 − x, that is, f (x) = 1 − x. However, not always we can take an equation with x and y and change it into a function. To see what can happen, consider some examples. We will look at several equations of the form F(x,y) = c. Formally, such an equation does not give a function (at least not immediately and not always), but it describes a certain set of points in the plane.

Example 1: Consider the equation   x + y = 1. The set of all points in plane that satisfy this equation is this straight line:

We used the known fact that equations of the form, say, ax + by = c give straight lines. As we observed above, the set given by this equation actually gives a curve that can be also interpreted as a graph of a certain function.

Example 2: Consider the equation   y² = x. The set of all points in plane that satisfy this equation is this:

To see this we used a simple trick, we "switched" the variables and imagined that x depends on y, then the equation describes the basic parabola, we just have to draw it properly (with switched axes). Again, we obtain a curve, but this time we cannot consider it a graph of some function. On the other hand, we can make it up out of two graphs. Solving the equality for y gives y = ±, which shows which two functions give the whole curve.

Example 3: Consider the equation

(|x| − 1)² + y² = 1.

To see this we used another trick. If x was in the equation without the absolute value, then the equation would describe a circle with radius 1 and center (1,0) (see below). Adding the absolute value to x means that also the symmetric (about the y-axis) shape satisfies this equation. Again, we cannot express this shape as one graph of a function, but we should be able to put it together as two graphs, the two hills above the x-axis as one and the two hills below as the other. Note that if we put in the equation the number on the right smaller than 1, we would get two circles that are apart, so it would not even be one uninterrupted curve.

Example 4: Consider the equation

(|x| − 1)² + y² = 0.

This example shows that an equation need not give a curve. Here we have two points, one can easily write an equation that describes even infinitely many points that are all isolated.

Example 5: Consider the equation

(|x| − 1)² + y² = −1.

Example 6: Consider the equation   sin(y) = x. The set of all points in plane that satisfy this equation is this:

This example is quite interesting. When we try to solve the given equation for y, we see that there are infinitely many solutions, functions   y_k = arcsin(x) + 2πk for all integers k satisfy this equation (see Trig functions in Functions - Theory - Elementary functions). Indeed, the picture correctly suggests that the set given by the given equation consists of infinitely many curves, each of which can be interpreted as a graph of a function.

We used these examples to show that an equation can define very strange sets in the plane, so it would be naive to expect that we can always make a function out of it. To get a better insight we look closer at what is usually needed.

Problem: We have an equation with x and y. We also have a point (x₀, y₀) that satisfies this equation. We want to find a neighborhood U of x₀ and a function y = y(x) on this neighborhood such that y(x₀) = y₀ (that is, the given point lies on the graph of this function) and for all x from U, the pairs (x, y(x)) satisfy the given equation (that is, the graph of this function actually agrees with the set given by the equation on a neighborhood of the given point).

Note that when investigating functions, we need some neighborhood to be able to find limits or a derivative, so it is natural to ask that we describe the curve as a function on a neighborhood of the given point.

Back to Example 2: If somebody gives us a point (4,2), then we can find a neighborhood of x₀ = 4, for instance U = (3,4.5) would do, and a function, namely y(x) = , which on U actually describes a part of the curve around the point (4,2).

In other words, locally, on a neighborhood of the given point, we were able to solve the given equation for y. Similarly, if somebody gives us a point (1,−1), then we can find a neighborhood of x₀ = 1, for instance U = (0,2) would do, and a function, namely y(x) = −, which on U actually describes a part of the curve around the point (1,−1).

Thus we can again locally express the curve as a function, but a different function then before. This is quite normal. Finally, consider the point (0,0). Here we are out of luck. No matter how small a neighborhood we take, there will always be points x for which we get two values on the curve, therefore we cannot express the curve around (0,0) as a function. This is, unfortunately, also quite normal. For instance, note that in Example 3 we can express the curve locally at all points with the exception of points (−2,0), (0,0), and (2,0).

This tells us the following. If we have a curve given by an equation, we often have a chance to express its parts around given points by functions (implicit functions), but sometimes this is not possible. Is there a way to somehow tell "good" points from "bad" points? In fact, there is, but it requires more advanced math (functions of more variables, partial derivatives). We will include the relevant statement here for the sake of completeness.

Theorem (Implicit function theorem).
Consider an equation F(x, y) = c. Let a point (x₀, y₀) satisfy this equation. If

then there exists a neighborhood U of x₀ and a function y = y(x) on this neighborhood such that y(x₀) = y₀ and F(x, y(x)) = c for all x from U.

Example: Consider the curve given by   y³ + 1 = xy. Does it define an implicit function on some neighborhood of (2,1)?

Solution: First we rearrange this equation to fit the above pattern:

xy − y³ = 1.

Now we find the necessary partial derivative.

Since this derivative is not zero, by the Implicit function theorem there is a desired function on some neighborhood of x₀ = 2. Unfortunately, we cannot find it reasonably easily. (There are formulas for roots of a cubic polynomial, but they are ugly and nobody remembers them anyway.)

What good is then all this business? First, we see that implicit curves are a richer family then the family of possible "nice" graphs of functions. In other words, we can easily describe implicitely objects that would be rather difficult to trace using graphs of functions (see below). That's the advantage of implicitly defined curves.

On the other hand, when we want to investigate these curves (for instance find tangent lines), we would much prefer to treat them as functions, since then we have very powerful tools. The theorem above tells us something, perhaps not too much, but note that tangent lines just happens to be a problem where local knowledge close to a given point is enough. What is even more interesting, there is a way to find derivative of an implicit function without really knowing its formula, just using the equation that defines it. This can be quite useful, see Implicit differentiation in Derivative - Theory - Implicit and parametric functions.

Two examples of working with implicit functions on the level we know now can be found in Solved Problems.

One last remark: We talked about changing implicit curves into functions. Can it be done the other way? The answer is positive and it is so simple that we are not going to waste another paragraph on it. If a curve is given as a graph of some function y = f (x), we can always express it by the implicit equation y − f (x) = 0. Of course, we can also do something different, say 2y − 2f (x) + 2 = 2, in fact we can express it in infinitely many different ways as an implicit curve. That's it, folks.

Bonus: Some famous curves.

Ellipse with semi-major axis A, semi-minor axis B, and center (a,b) is given by the equation

Obviously, taking A = B = r we get a circle.

Parametric functions
Back to Theory - Implicit and parametric functions