Range (mathematics)

FORMAL DEFINITION

Given a Function

f\colon A
ightarrow B

, the range of

f

, is defined to be the Set
:

\{ x \in B : x = f(a) \mbox{ for some } a \in A \}.

The range of ''f'' is sometimes denoted ran(''f'').

The range should not be confused with the Codomain ''B''. The range is a Subset of the codomain, but is not necessarily equal to the codomain, since there may be elements of the codomain which are not elements of the range. The codomain is sometimes taken to be the range, but more often is some standard set, such as the real numbers or the complex numbers, which contains the range. (Older books sometimes call what is now called the codomain the range, and what is now called the range the image set.) A function whose range equals its codomain is called Onto or surjective.

EXAMPLES

Let the function f be a function on the Real Number s:

:

f\colon \mathbb{R}
ightarrow\mathbb{R}

defined by

:

f(x) = x^2

The codomain of f is R, and ''f'' takes all Nonnegative values but never takes Negative values, and thus the range is in fact the set R⁺—non-negative reals, i.e. the Interval [0,∞):

:

0\leq f(x)<\infty.

Now let g be a function on the Real Number s:

:

g\colon \mathbb{R}
ightarrow\mathbb{R}

defined by

:

g(x) = 2x

In this case the image of ''g'' equals R, its codomain, since, for any real number ''y'',

:

g(y/2) = y.

In other words, ''g'' is onto R.

SEE ALSO

Codomain

Domain (mathematics)

Bijection, Injection And Surjection

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Range (mathematics)