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Through[p[f,g,…][x,y,…]]

gives p[f[x,y,…],g[x,y,…],…].

Through[expr,h]

performs the transformation wherever h occurs in the head of expr.

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Through

Through[p[f,g,…][x,y,…]]

gives p[f[x,y,…],g[x,y,…],…].

Through[expr,h]

performs the transformation wherever h occurs in the head of expr.

Details

  • Through distributes operators that appear inside the heads of expressions.
  • Parallelize[Through[p[f,g,…][x,y,…]]] computes Through[p[f,g,…][x,y,…]] in parallel on all subkernels. »

Examples

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Basic Examples  (2)

Apply all functions in a list to the same argument:

Wolfram Language code: Through[{f, g, h}[x]]

Apply a sum of functions by forming the sum of the individual applications:

Wolfram Language code: Through[(f + g + h)[x, y]]

Scope  (2)

By default, the operation is performed for any function in the head:

Wolfram Language code: Through[(f + g + h)[x, y]]

Perform the operation only if the head is a sum of terms:

Wolfram Language code: Through[(f + g + h)[x, y], Plus]
Wolfram Language code: Through[(f g)[x, y], Plus]

Applications  (2)

Apply a differential operator:

Wolfram Language code: ((D[#, x]&) + (D[#, x, x]&))[f[x]]
Wolfram Language code: Through[%]
Wolfram Language code: NestList[Through, f[a][b][c][d], 3]

Properties & Relations  (5)

Wolfram Language code: Framed[x][y]
Wolfram Language code: Through[%]

Tabulate square roots of values without using auxiliary variables:

Wolfram Language code: TableForm[Composition[Through, {Identity, Sqrt}] /@ {0, 1.0, 2.0, 3.0, 4.0}]

Compute Through in parallel:

Wolfram Language code: Parallelize[Through[{f, g, h}[a, b]]]

Through[p[f,g,…][x]] is equivalent to Comap[p[f,g,…],x]:

Wolfram Language code: Through[{f, g, h}[x]]
Wolfram Language code: Comap[{f, g, h}, x]

Compare this to the operator form of Comap:

Wolfram Language code: Comap[{f, g, h}][x]

Through[p[f,g,…][x,y,…]] is equivalent to ComapApply[p[f,g,…],{x,y,…}]:

Wolfram Language code: Through[{f, g, h}[x, y, z]]
Wolfram Language code: ComapApply[{f, g, h}, {x, y, z}]

Compare this to the operator form of ComapApply:

Wolfram Language code: ComapApply[{f, g, h}][{x, y, z}]
Wolfram Research (1988), Through, Wolfram Language function, https://reference.wolfram.com/language/ref/Through.html.

Text

Wolfram Research (1988), Through, Wolfram Language function, https://reference.wolfram.com/language/ref/Through.html.

CMS

Wolfram Language. 1988. "Through." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Through.html.

APA

Wolfram Language. (1988). Through. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Through.html

BibTeX

@misc{reference.wolfram_2026_through, author="Wolfram Research", title="{Through}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Through.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_through, organization={Wolfram Research}, title={Through}, year={1988}, url={https://reference.wolfram.com/language/ref/Through.html}, note=[Accessed: 13-June-2026]}