This is documentation for Mathematica 4, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.1)
Wolfram Research, Inc.

Expressions with Heads That Are Not SymbolsStructural Operations

2.2.9 Advanced Topic: 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.

Some functional operations.

This represents the composition of the functions f, g and h.

In[1]:= Composition[f, g, h]

Out[1]=

You can manipulate compositions of functions symbolically.

In[2]:= InverseFunction[Composition[%, q]]

Out[2]=

The composition is evaluated explicitly when you supply a specific argument.

In[3]:= %[x]

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.

In[4]:= (f + g)[x]

Out[4]=

Using Through, you can convert the expression to a more explicit form.

In[5]:= Through[%, Plus]

Out[5]=

This corresponds to the mathematical operator .

In[6]:= Identity + (D[#, x]&)

Out[6]=

Mathematica does not automatically apply the separate pieces of the operator to an expression.

In[7]:= % [x^2]

Out[7]=

You can use Through to apply the operator.

In[8]:= Through[%, Plus]

Out[8]=

Operations for working with operators.

This has a complicated expression as a head.

In[9]:= t = ((1 + a)(1 + b))[x]

Out[9]=

Functions like Expand do not automatically go inside heads of expressions.

In[10]:= Expand[%]

Out[10]=

With the Heads option set to True, MapAll goes inside heads.

In[11]:= MapAll[Expand, t, Heads->True]

Out[11]=

The replacement operator /. does go inside heads of expressions.

In[12]:= t /. a->1

Out[12]=

You can use Operate to apply a function specifically to the head of an expression.

In[13]:= Operate[p, t]

Out[13]=

Expressions with Heads That Are Not SymbolsStructural Operations