Range of a function

The range of a function is the set of output values it actually attains over its domain, distinct from the larger set of possible outputs called its codomain.

Function: A relation that assigns each element of a domain exactly one element of a codomain. A function’s rule and domain determine which outputs can enter its range.
Inverse function: A function that reverses the input-output assignments of a bijective function. Solving for inputs in terms of outputs can reveal which output values are attainable.
Quadratic function: A polynomial function of degree two, whose graph is a parabola. Its vertex gives the attained minimum or maximum that bounds its range.
Preimage: The set of inputs that a function maps to a specified output or set of outputs. Preimages ask which inputs produce chosen outputs, reversing the range question.
Domain of a function: The set of inputs for which a function is defined. Changing the allowed inputs can change which values the function attains.
Critical point: A point in a function’s domain where its derivative is zero or undefined. For differentiable functions, critical points help locate extreme values that bound the range.
Absolute value function: A function such as f(x)=|x| that maps real inputs to their nonnegative magnitudes. Its range is bounded below by zero, which the function attains.
Restricted function: A function formed by limiting the domain of another function while retaining its rule. Restricting the domain can remove outputs from the original range.
Codomain: The specified set in which a function’s outputs are required to lie. The range is contained in the codomain but need not equal it.
Global maximum and minimum: The greatest and least values a function attains over its domain, when they exist. Attained extrema often set the endpoints of a function’s range.

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