AP Precalculus - Foundational Concepts

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Function

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A mathematical relation that maps a set of input values (domain) to a set of output values (range) such that each input value is mapped to exactly one output value. Functions can be represented graphically, numerically, analytically, or verbally.

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Domain

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The set of all possible input values (x-values) for which a function is defined. When finding domain, check for restrictions: denominators cannot equal zero, and radicands of even roots must be non-negative.

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Range

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The set of all possible output values (y-values) that a function can produce. Range can be determined from a graph by identifying all y-values the graph attains, or analytically by considering the function's behavior.

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Increasing Function

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A function is increasing over an interval if, as input values increase, output values always increase. Formally: for all a and b in the interval, if a < b, then f(a) < f(b). On a graph, the curve rises from left to right.

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Decreasing Function

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A function is decreasing over an interval if, as input values increase, output values always decrease. Formally: for all a and b in the interval, if a < b, then f(a) > f(b). On a graph, the curve falls from left to right.