4.4 Implicitly Defined Functions

Implicit vs. explicit relations

An explicit function is written as y = f(x), directly giving y in terms of x. An implicitly defined function relates x and y in an equation, like x² + y² = 25, without isolating y. These equations may not pass the vertical line test, so they can define more than one y-value for a single x, meaning the relation isn't always a function.

Analyzing implicit relations

Graphs of implicit relations, such as conics (circles, ellipses) or curves like x² + y³ = xy, can be analyzed for symmetry, intercepts, and behavior by substituting values or testing (-x, -y), (-x, y), (x, -y) into the equation. Symmetry about the x-axis, y-axis, or origin can be identified this way without solving for y explicitly.

Using technology and reasoning

Since solving implicit equations algebraically for y can be difficult or impossible, graphing technology is often used to visualize the relation. Key features—like where the graph crosses axes, has maximum/minimum x or y values, or exhibits symmetry—can be determined by reasoning about the implicit equation itself, not just from a formula.

Key terms

Implicitly defined function

A relation between x and y expressed in an equation without isolating y, e.g., x² + y² = 25.

Explicit function

A function written directly as y = f(x).

Vertical line test

A test where, if any vertical line crosses a graph more than once, the relation is not a function.

Symmetry about the x-axis

Occurs when replacing y with -y produces an equivalent equation.

Why use graphing technology for implicit equations?

Because many implicit equations can't be easily or algebraically solved for y, so technology helps visualize the graph and key features.

Try an AP-style question

Why is x² + y² = 25 not the graph of a function of x?

1. Some x-values have two y-values
2. It isn't continuous on its domain
3. It has no x-intercepts at all
4. It passes through the origin twice