1.31 Function Notation and Set Definitions

Function Notation and Set Definitions establishes the vocabulary for the compact symbolic system used to name functions and reference their outputs, along with the two sets of values that bound a function's behavior—the complete collection of permitted inputs and the complete collection of resulting outputs—and the roles those inputs and outputs play in describing a relationship of dependence.

Function Notation

Function notation is the symbolic convention of naming a function with a letter, most commonly f, and writing f(x) to represent the output produced when x is supplied as input, read as "f of x." Function notation replaces the more cumbersome y = ... form with a labeled form that makes explicit which function produced a given output and which input produced it.

Function Value

The function value is the specific output produced when a particular input is substituted into a function written in function notation, obtained by replacing every occurrence of the input variable with the given input value and evaluating the resulting expression. For the function f(x) = 2x + 3, the function value at x = 4, written f(4), is found by substitution: f(4) = 2(4) + 3 = 11.

Function Domain

The function domain is the complete set of all permissible input values for which a function is defined and produces a valid output. For most elementary algebraic functions, the domain consists of every real number, unless a specific restriction—such as a denominator that cannot equal zero, or a radicand that cannot be negative—excludes certain values from being valid inputs.

Function Range

The function range is the complete set of all output values that a function actually produces across its entire domain. The range describes every value the function can output, and depending on the function's structure, it may be limited to a smaller set of values than the entire set of real numbers, even when the domain itself is unrestricted.

Independent Variable

The independent variable is the variable representing the input of a function, whose value may be freely chosen from within the function's domain without depending on any other quantity in the relationship. In the function f(x) = 2x + 3, x is the independent variable, since its value can be selected freely before the function determines a corresponding output.

Dependent Variable

The dependent variable is the variable representing the output of a function, whose value is determined by, and therefore depends on, the value chosen for the independent variable. Writing y = f(x) designates y as the dependent variable, since its value is fixed only once a specific value has been assigned to the independent variable x.

Together, these definitions establish function notation as the compact system for naming and evaluating functions, and describe how a function's domain and range bound its permissible inputs and resulting outputs, while the independent and dependent variable roles clarify which quantity is freely chosen and which quantity results from that choice.