# Applying Functions to Parts of Expressions

If you have a list of elements, it is often important to be able to apply a function separately to each of the elements. You can do this in the Wolfram Language using Map.

This applies f separately to each element in a list:
 In:= Out= This defines a function which takes the first two elements from a list:
 In:= You can use Map to apply take2 to each element of a list:
 In:= Out= Map[f,{a,b,…}] apply f to each element in a list, giving {f[a],f[b],…}

Applying a function to each element in a list.

What Map[f,expr] effectively does is to "wrap" the function f around each element of the expression expr. You can use Map on any expression, not just a list.

This applies f to each element in the sum:
 In:= Out= This applies Sqrt to each argument of g:
 In:= Out= Map[f,expr] applies f to the first level of parts in expr. You can use MapAll[f,expr] to apply f to all the parts of expr.

This defines a 2x2 matrix m:
 In:= Out= Map applies f to the first level of m, in this case the rows of the matrix:
 In:= Out= MapAll applies f at all levels in m. If you look carefully at this expression, you will see an f wrapped around every part:
 In:= Out= In general, you can use level specifications as described in "Levels in Expressions" to tell Map to which parts of an expression to apply your function.

This applies f only to the parts of m at level 2:
 In:= Out= Setting the option wraps f around the head of each part, as well as its elements:
 In:= Out= Map[f,expr] or f/@expr apply f to the first‐level parts of expr MapAll[f,expr] or f//@expr apply f to all parts of expr Map[f,expr,lev] apply f to each part of expr at levels specified by lev

Ways to apply a function to different parts of expressions.

Level specifications allow you to tell Map to which levels of parts in an expression you want a function applied. With MapAt, however, you can instead give an explicit list of parts where you want a function applied. You specify each part by giving its indices, as discussed in "Parts of Expressions".

Here is a 2x3 matrix:
 In:= Out= This applies f to parts {1,2} and {2,3}:
 In:= Out= This gives a list of the positions at which b occurs in mm:
 In:= Out= You can feed the list of positions you get from Position directly into MapAt:
 In:= Out= To avoid ambiguity, you must put each part specification in a list, even when it involves only one index:
 In:= Out= MapAt[f,expr,{part1,part2,…}] apply f to specified parts of expr

Applying a function to specific parts of an expression.

Here is an expression:
 In:= Out= This is the full form of t:
 In:= Out//FullForm= You can use MapAt on any expression. Remember that parts are numbered on the basis of the full forms of expressions:
 In:= Out= MapIndexed[f,expr] apply f to the elements of an expression, giving the part specification of each element as a second argument to f MapIndexed[f,expr,lev] apply f to parts at specified levels, giving the list of indices for each part as a second argument to f

Applying a function to parts and their indices.

This applies f to each element in a list, giving the index of the element as a second argument to f:
 In:= Out= This applies f to both levels in a matrix:
 In:= Out= Map allows you to apply a function of one argument to parts of an expression. Sometimes, however, you may instead want to apply a function of several arguments to corresponding parts of several different expressions. You can do this using MapThread.

 MapThread[f,{expr1,expr2,…}] apply f to corresponding elements in each of the expri MapThread[f,{expr1,expr2,…},lev] apply f to parts of the expri at the specified level

Applying a function to several expressions at once.

This applies f to corresponding pairs of list elements:
 In:= Out= MapThread works with any number of expressions, so long as they have the same structure:
 In:= Out= Functions like Map allow you to create expressions with parts modified. Sometimes you simply want to go through an expression, and apply a particular function to some parts of it, without building a new expression. A typical case is when the function you apply has certain side effects, such as making assignments, or generating output.

 Scan[f,expr] evaluate f applied to each element of expr in turn Scan[f,expr,lev] evaluate f applied to parts of expr on levels specified by lev

Evaluating functions on parts of expressions.

Map constructs a new list in which f has been applied to each element of the list:
 In:= Out= Scan evaluates the result of applying a function to each element, but does not construct a new expression:
 In:=    Scan visits the parts of an expression in a depthfirst walk, with the leaves visited first:
 In:=     