Explicit vs Implicit Functions
When you first learn about functions, they look something like this:
$$y = x^2 \qquad y = \sin(x) \qquad y = \sqrt{1 - x^2}$$
Each formula tells you exactly what $y$ is for any given $x$. These are called explicit functions. But there is a whole family of curves that no explicit formula can describe — and to reach them, we need implicit functions.
Explicit Functions
A function is explicit when $y$ is isolated on one side:
$$y = f(x)$$
You plug in an $x$, and the formula hands you back exactly one $y$. This is the "one output per input" rule — the vertical line test. Plotting explicit functions is straightforward: sweep $x$ across a range, compute $y$ at each point, and connect the dots.
Here $y = x^2$ is explicit. For every value of $x$, there is exactly one value of $y$.
The Limitation
The one-output rule is also a constraint. Consider the unit circle — all points at distance 1 from the origin:
$$x^2 + y^2 = 1$$
Can we write $y$ explicitly in terms of $x$? Solving for $y$ gives:
$$y = \pm\sqrt{1 - x^2}$$
The $\pm$ is the problem. For any $x$ between $-1$ and $1$, there are two values of $y$ — one on the upper arc and one on the lower arc. No single explicit function can produce both. We are forced to split the circle into two halves and describe each separately:
$$y = \sqrt{1 - x^2} \quad \text{(upper semicircle)}$$ $$y = -\sqrt{1 - x^2} \quad \text{(lower semicircle)}$$
Any closed curve — circles, ellipses, loops — runs into this same wall. Explicit functions simply cannot fold back over themselves.
Implicit Functions
An implicit function does not isolate $y$. Instead, we write a condition that $x$ and $y$ must satisfy together:
$$F(x, y) = 0$$
The curve is the zero set of $F$ — every point $(x, y)$ where this condition holds. The circle becomes:
$$F(x, y) = x^2 + y^2 - 1 = 0$$
Now both arcs are captured in a single equation, with no $\pm$ anywhere.
The implicit form describes the full circle at once — something impossible for any single explicit function.
Side-by-Side Comparison
| Explicit | Implicit | |
|---|---|---|
| Form | $y = f(x)$ | $F(x, y) = 0$ |
| Outputs per input | Exactly one | Any number |
| Closed curves | Not possible | Natural |
| Plotting | Sweep $x$, compute $y$ | Check every point |
The cost of the extra generality is that implicit curves are harder to plot. You can no longer just sweep $x$ — you must sample points across the entire plane and check whether $F(x, y) \approx 0$. This is why implicit curve plotters use techniques like marching squares, which trace the zero set directly.
An Implicit Curve That Cannot Be Made Explicit
The lemniscate of Bernoulli is a figure-eight shaped curve defined by:
$$(x^2 + y^2)^2 = 2(x^2 - y^2)$$
Rearranging to the standard implicit form:
$$(x^2 + y^2)^2 - 2(x^2 - y^2) = 0$$
The figure-eight crosses itself at the origin. No explicit function can self-intersect, so there is no way to write this curve as $y = f(x)$. The implicit form handles it effortlessly.
Implicit Does Not Mean Unknown
The word "implicit" might suggest something hidden or vague, but the curve is perfectly well-defined — every point satisfying $F(x, y) = 0$ belongs to it. What is implicit is the relationship between $x$ and $y$, not the curve itself.
In fact, the Implicit Function Theorem tells us that in smooth regions — away from singular points — an implicit curve locally looks like an explicit function. Near most points, you could in principle solve for $y$ in terms of $x$. It is only globally, across the whole curve, that a single explicit formula breaks down.
Conclusion
Explicit functions are the familiar workhorses: easy to evaluate, easy to plot, and perfectly sufficient for many purposes. But the moment a curve loops back, crosses itself, or forms a closed shape, explicit functions hit their limit. Implicit functions, defined by a condition $F(x, y) = 0$, describe any curve in the plane — no matter how complex its shape. The tradeoff is that they are harder to plot but far more expressive.