Range of a Function

Definition

In mathematics, the range of a function may refer to either of two closely related concepts:

• The codomain of the function
• The image of the function

Given two sets X and Y, a binary relation f between X and Y is a (total) function (from X to Y) if for every x in X there is exactly one y in Y such that f relates x to y. The sets X and Y are called domain and codomain of f, respectively. The image of f is then the subset of Y consisting of only those elements y of Y such that there is at least one x in X with f(x) = y.

In algebra, the range (or codomain) of a function is all of the possible outputs of the function. That is, if x is any element of the domain of some function f, then f(x) is in the range of the function f.

Examples:

• Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} be a set of ordered pairs such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = \{(1,2), (2,3), (4, 7), (13, 9), (-20, 0)\}} . The range is the set of all y values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , so the range is $\{0,2,3,7,9\}$.
• The range of $g(x)=1/x$ is all real numbers EXCEPT for 0. We know this because for all nonzero real numbers M, 1/M is a nonzero number and is in the domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) } (since the domain of this function is all nonzero numbers). So, we know that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/(1/M) = M } is in the range, where M is all nonzero numbers. There is no real number M such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/M = 0 } though, which is why 0 is not in the range of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) } .
• The range of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x) = x^2 + 2 } is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [2,\inf) } . We can see this on the graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x) } easily: the lowest point, or vertex, of the parabola is at (0, 2), so 2 is in the range. The parabola extends up to infinity on either side of the vertex, so we know that the range must be all numbers from 2 to infinity.

Resources

• Domain and Range, Interactive Mathematics
• Domain and Range: Basic Idea, patrickJMT
• Finding Domain and Range with Graphs, Lumen Learning
• Finding Domain and Range of a Piecewise Function, patrickJMT
• How to Find Range + Example Problems, Math Culus