AP® Precalculus · Unit 4 · Topic 4.5
Implicitly Defined Functions
An implicit equation relates x and y without isolating either one, and often describes a curve that fails the one-output-per-input rule, so you split it into separate pieces that are each functions.
Most equations you graph, like y = x2 + 1, already tell you how to build y from x. An implicit equation, like x2 + y2 = 25, just states a relationship both variables must satisfy, with no variable singled out. That relationship can trace a shape, like a circle, that a single function could never produce, because a function must give exactly one output for every input.
To work with an implicit equation, you solve it for y in terms of x, which usually forces you to split the curve into pieces. For x2 + y2 = 25, isolate y2 = 25 - x2, then take square roots: y = √(25 - x2) or y = -√(25 - x2). Each piece alone passes the vertical line test. The positive branch traces the top half of the circle, the negative branch traces the bottom half. Together they rebuild the whole circle, but separately, each is a legitimate function with its own domain, here -5 ≤ x ≤ 5.
This splitting habit carries forward into everything else involving curves defined by relationships rather than direct formulas, including conic sections and parametric motion later in the unit. Recognizing when an equation is implicit, and knowing you can often coax explicit functions out of it piece by piece, is the same move you will make whenever a graph bends back on itself and refuses to be one tidy formula.
Key terms
- implicit equation
- An equation relating x and y without solving for either variable explicitly.
- explicit function
- A function written as y equals some expression in x alone, like y = √(25 - x2).
- branch
- One piece of a curve obtained by solving an implicit equation, that by itself satisfies the function rule.
- vertical line test
- A graph represents a function only if every vertical line crosses it at most once.
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End of topic.