Range of a Function
Functions & Advanced AlgebraThe range of a function is the set of all possible output values produced by the function.
Definition
The range is the set of all possible outputs (answers) a function can produce. After you plug in every allowed input, the range is every output you could ever get.
Example
If $f(x) = x^2$, the outputs are always $0$ or positive. You can never get a negative number out of squaring. So the range is all numbers greater than or equal to $0$.
Key Insight
While the domain is the "inputs allowed," the range is the "outputs actually produced." Not every number in the codomain has to show up as an output.
Definition
The range of a function is the set of all $y$-values (outputs) that result from substituting every value in the domain into the function. To find the range, determine what output values are actually achievable.
Example
For $f(x) = 2x + 1$ on all real numbers, every real $y$ is achievable (solve $y = 2x + 1$ for $x$). Range: all reals. For $g(x) = x^2$, no negative $y$ is achievable. Range: $[0, \infty)$.
Key Insight
The range is sometimes called the image of the function. It is a subset of the codomain. Distinguishing range from codomain matters in determining whether a function is surjective (onto).
Definition
The range (or image) of $f: A \to B$ is the set $f(A) = \{f(a) : a \in A\}$, a subset of the codomain $B$. A function is surjective if and only if its range equals its codomain. Finding the range analytically may require solving for $x$ in terms of $y$ and determining which $y$-values yield solutions.
Example
For $f(x) = (x^2 - 1)/(x^2 + 1)$, note $-1 < f(x) < 1$ for all real $x$, so the range is $(-1, 1)$. This requires bounding analysis, not just algebraic inversion.
Key Insight
For measurable functions in real analysis, the range interacts with measure theory: preimages of measurable sets must be measurable for the function to be measurable.