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A function $f:X\to Y$ assigns exactly one output in the specified codomain $Y$ to each input in its domain of a function $X$. Its rule may be a formula, a finite table, or another unambiguous assignment. The outputs actually reached form its image of a function $f(X)\subseteq Y$. A codomain states where outputs are allowed; the image of a function records where they actually land. Oscar Levin's introduction makes this distinction explicit. ^[ref].

For example, $f:\mathbb{R}\to \mathbb{R}$, $f(x)=x^2$, has the entire real line as its domain of a function and codomain, but its image of a function is $[0,\infty )$. Both $-2$ and $2$ give the output $4$: several inputs sharing an output is permitted. Assigning two distinct outputs to one input is not permitted. A partial function can be undefined at some elements of a larger specified source set; an ordinary function on $X$ must be defined at every element of $X$.

Three properties describe how a function uses its target. An injective function never sends distinct inputs to the same output. A surjective function reaches every member of its specified codomain. A bijection does both, pairing each source element with exactly one target element. These are different requirements, as finite arrow diagrams show.

The chosen codomain matters even when the formula does not change. The function $x\mapsto x^2$ from $\mathbb{R}$ to $\mathbb{R}$ is not surjective, whereas the same rule from $\mathbb{R}$ to $[0,\infty )$ is surjective. Both fail injectivity. Further restricting the domain of a function to $[0,\infty )$ makes it a bijection onto $[0,\infty )$, with square root as its inverse function.

Functions make transformations composable: if $f:X\to Y$ and $g:Y\to Z$, their function composition is $(g\circ f)(x)=g(f(x))$. Identifying each intermediate codomain prevents attempting to feed an output into a rule that is not defined there. When a transformation is a bijection, an inverse function recovers the original input without ambiguity. This is useful whenever information must be transported or a calculation reversed.

A functional is a function whose inputs are themselves functions, curves, or other elements of a function space.

Functional analysis brings linear algebra and topology together to study vector spaces of functions and the linear operators acting on them. A function space such as $C([0,1])$ has infinitely many independent directions: the monomials $1,x,x^2,\dots$ are linearly independent. Finite linear combinations alone cannot describe all its continuous functions. A useful theory must also say when approximations converge, whether their limits belong to the chosen space, and which operations preserve that norm convergence. Normed vector spaces, Banach spaces, and bounded linear operators provide these three controls.^[ref]

A bounded linear operator $T:X\to Y$ between normed vector spaces satisfies $\Vert Tf{\Vert }_Y\le M\Vert f{\Vert }_X$ for every $f$, with one finite constant $M$. The least possible constant is its operator norm. By linearity, the same estimate controls approximation errors:This is exactly continuity for a linear map between normed vector spaces; neither space needs to be complete. It turns an input accuracy guarantee into an output accuracy guarantee, even when the input has infinitely many degrees of freedom.^[ref]

For a concrete linear functional on $C([0,1])$, take the integral $F(f)={\int }_0^{1}f(x)dx$. In the supremum norm,The constant function $f=1$ attains equality, so $\Vert F\Vert =1$. For $f(x)=x^2$, the output is $F(f)=1/3$; any approximation $g$ with $\Vert g-x^2{\Vert }_{\infty }\le 0.01$ therefore has $|F(g)-1/3|\le 0.01$. The Cauchy-Schwarz inequality also gives $|F(f)|\le \Vert f{\Vert }_2\Vert 1{\Vert }_2=\Vert f{\Vert }_2$ on this interval, again with equality at $1$. Thus integration remains a bounded linear functional under the L2 norm, although point evaluation behaves very differently.

The beauty of this viewpoint is that one estimate can justify an entire approximation procedure. Once a sequence converges in the chosen norm, every bounded linear operator carries its limit along. Completeness ensures that Cauchy approximations actually describe an element of the space. The choice of norm matters: it determines which errors are small and which measurements are stable. The examples below distinguish an average, which ignores sufficiently narrow peaks, from point evaluation, which detects their full height. For applications, this decides whether reducing an overall fitting error guarantees an accurate reported measurement. A small L2 norm of the error controls its integral on this interval, while it gives no such guarantee for a single sampled value.

A functional equation specifies a function through an identity involving values of that function at several arguments.

A function class is a set whose elements are functions, usually sharing a common domain of a function and codomain.

A function collision is a pair of distinct inputs $x\ne x^{\prime }$ with $f(x)=f(x^{\prime })$. An injective function has no collisions. For a two-to-one function every nonempty fiber has two elements and hence one unordered collision pair. Finding such a pair through an oracle is the task of quantum collision finding.

In the finite example under image and preimage of a function, the pair $\{a,b\}$ is a function collision: both inputs give zero. The function retains their shared output but loses the distinction between them. If a finite set has more inputs than the number of available outputs, the pigeonhole principle forces a function collision. Equal source and target cardinalities alone do not prevent one.

For a finite function, let $m_y$ be the cardinality of its fiber of a function over $y$. The number of unordered function collisions is ${\sum }_y\left(\genfrac{}{}{0ex}{}{m_y}{2}\right)$, because each pair must lie in one common fiber. The example's nonempty fiber sizes are $2,1,1$, giving exactly one collision pair.

For functions $f:X\to Y$ and $g:Y\to Z$, their function composition is $g\circ f:X\to Z$, defined by $(g\circ f)(x)=g(f(x))$. The rightmost function is applied first. Matching the intermediate codomain and domain of a function makes the expression well-defined.

For self-maps of the real numbers, take $f(x)=x+1$ and $g(x)=2x$. Then $(g\circ f)(x)=2x+2$, whereas $(f\circ g)(x)=2x+1$. Thus function composition need not be commutative. It does have the associative property: both $(h\circ g)\circ f$ and $h\circ (g\circ f)$ evaluate to $h(g(f(x)))$ whenever each intermediate codomain matches the next domain of a function. The identity function leaves composition unchanged, making self-maps particularly useful as a monoid of transformations.

A function in extension is a function specified by its domain and the value at each input in that domain. Distinct functions in intension can have this same extension. Properties of the extension of an arbitrary program are semantic properties, to which Rice theorem applies when the property is nontrivial.

A function input to a function $f:X\to Y$ is an element $x$ of its domain of a function $X$, also called its argument. The corresponding function output is $f(x)$.

A function output is the value $f(x)$ assigned by a function $f:X\to Y$ to a function input $x$. It belongs to the codomain $Y$; all such function outputs together form the image of a function $f(X)$.

A function space is a collection of functions with common domain and codomain, often supplied with a vector space structure and a norm or topology. Examples include L2 spaces and Sobolev spaces.

A functor assigns an object of a category $F(A)$ to each object $A$ and a categorical morphism $F(f):F(A)\to F(B)$ to each arrow $f:A\to B$. It preserves the rules for composing arrows:For example, the forgetful functor from the category of groups to the Category of sets keeps the underlying sets and maps of elements while forgetting the group operations. A group homomorphism remains a function, and composing the underlying functions gives the underlying map of the composite group homomorphism.

Another example sends a set $X$ to its power set $\mathcal{P}(X)$. It sends a function $f:X\to Y$ to the function $\mathcal{P}(f):\mathcal{P}(X)\to \mathcal{P}(Y)$ defined by $S\mapsto f(S)$, the image of the subset $S$. For $f(x)=x+1$ and $g(x)=2x$, the subset $\{0,2\}$ goes to $\{1,3\}$ and then $\{2,6\}$; applying the composite $g\circ f$ directly also gives $\{2,6\}$. The equality of these routes expresses preservation of composition in a category.