1.32 Function Representation Definitions

Function Representation Definitions establishes the vocabulary for the different formats in which a single function can be displayed—as a table of paired values, as a symbolic rule, or as a graph—and describes the principle that guarantees these different formats, despite their different appearances, all describe the exact same underlying function.

Function Table

A function table is a representation of a function as an organized list of input values paired with their corresponding output values, typically arranged in two columns or rows, one for inputs and one for outputs. A function table displays only a finite sample of a function's input-output pairs, even when the function itself is defined over an infinite domain, making it a useful but necessarily incomplete snapshot of the function's behavior.

Function Rule

A function rule is a representation of a function as a symbolic expression or equation, most commonly written in function notation, that specifies exactly how to compute the output for any given input. The function rule f(x) = 2x + 3 fully determines the function's behavior for every possible input, in contrast to a function table, which lists only specific input-output pairs individually.

Function Graph

A function graph is a representation of a function as a set of plotted points on the coordinate plane, where each point's horizontal coordinate is an input value and its vertical coordinate is the corresponding output value. The function graph provides a visual depiction of how a function's output changes as its input changes, making overall patterns of behavior—such as increasing, decreasing, or constant trends—directly visible.

Representation Equivalence

Representation equivalence is the principle that a function table, a function rule, and a function graph, when constructed from the same underlying function, describe exactly the same relationship between inputs and outputs, differing only in format rather than in meaning. Any specific input-output pair appearing in a function table must also satisfy the corresponding function rule and must correspond to a point lying on the function's graph; a discrepancy among the three representations indicates an error in one of them rather than a genuine difference in the function itself.

Together, these definitions describe the three standard formats used to display a function—table, rule, and graph—and establish representation equivalence as the guiding principle that ties them together, allowing information about a function to be translated freely between whichever representation is most useful for a given purpose.