This is documentation for Mathematica 6, which was
based on an earlier version of the Wolfram Language.
 Mathematica Tutorial Functions »|Tutorials »

# Working with Operators

You can think of an expression like f[x] as being formed by applying an operator f to the expression x. You can think of an expression like f[g[x]] as the result of composing the operators f and g, and applying the result to x.
 Composition[f,g,...] the composition of functions f, g, ... InverseFunction[f] the inverse of a function f Identity the identity function

Some functional operations.

This represents the composition of the functions f, g and h.
 Out[1]=
You can manipulate compositions of functions symbolically.
 Out[2]=
The composition is evaluated explicitly when you supply a specific argument.
 Out[3]=
You can get the sum of two expressions in Mathematica just by typing x+y. Sometimes it is also worthwhile to consider performing operations like addition on operators.
You can think of this as containing a sum of two operators f and g.
 Out[4]=
Using Through, you can convert the expression to a more explicit form.
 Out[5]=
This corresponds to the mathematical operator .
 Out[6]=
Mathematica does not automatically apply the separate pieces of the operator to an expression.
 Out[7]=
You can use Through to apply the operator.
 Out[8]=
 Identity[expr] the identity function Through[p[f1,f2][x],q] give p[f1[x], f2[x]] if p is the same as q Operate[p,f[x]] give p[f][x] Operate[p,f[x],n] apply p at level n in f MapAll[p,expr,Heads->True] apply p to all parts of expr, including heads

Operations for working with operators.

This has a complicated expression as a head.
 Out[9]=
Functions like Expand do not automatically go inside heads of expressions.
 Out[10]=