f is a function from
domain X to
codomain Y. The smaller oval inside
Y is the
image of
f. Sometimes "range" refers to the image and sometimes to the codomain.
In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage. Modern usage almost always uses range to mean image.
The codomain of a function is some arbitrary set. In real analysis, it is the real numbers. In complex analysis, it is the complex numbers.
The image of a function is the set of all outputs of the function. The image is always a subset of the codomain.
Contents

Distinguishing between the two uses 1

Formal definition 2

See also 3

Notes 4

References 5
Distinguishing between the two uses
As the term "range" can have different meanings, it is considered a good practice to define it the first time it is used in a textbook or article.
Older books, when they use the word "range", tend to use it to mean what is now called the codomain.^{[1]}^{[2]} More modern books, if they use the word "range" at all, generally use it to mean what is now called the image.^{[3]} To avoid any confusion, a number of modern books don't use the word "range" at all.^{[4]}
As an example of the two different usages, consider the function f(x) = x^2 as it is used in real analysis, that is, as a function that inputs a real number and outputs its square. In this case, its codomain is the set of real numbers \mathbb{R}, but its image is the set of nonnegative real numbers \mathbb{R}^+, since x^2 is never negative if x is real. For this function, if we use "range" to mean codomain, it refers to \mathbb{R}. When we use "range" to mean image, it refers to \mathbb{R}^+.
As an example where the range equals the codomain, consider the function f(x) = 2x, which inputs a real number and outputs its double. For this function, the codomain and the image are the same (the function is a surjection), so the word range is unambiguous; it is the set of all real numbers.
Formal definition
When "range" is used to mean "codomain", the range of a function must be specified. It is often assumed to be the set of all real numbers, and {y  there exists an x in the domain of f such that y = f(x)} is called the image of f.
When "range" is used to mean "image", the range of a function f is {y  there exists an x in the domain of f such that y = f(x)}. In this case, the codomain of f must be specified, but is often assumed to be the set of all real numbers.
In both cases, image f ⊆ range f ⊆ codomain f, with at least one of the containments being equality.
See also
Notes

^ Hungerford 1974, page 3.

^ Childs 1990, page 140.

^ Dummit and Foote 2004, page 2.

^ Rudin 1991, page 99.
References

Childs (2009). A Concrete Introduction to Higher Algebra. Undergraduate Texts in Mathematics (3rd ed.). Springer.

Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley.

Hungerford, Thomas W. (1974). Algebra. Graduate Texts in Mathematics 73. Springer.

Rudin, Walter (1991). Functional Analysis (2nd ed.). McGraw Hill.
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