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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
Identitythe identity function

Some functional operations.

This represents the composition of the functions f, g and h.
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You can manipulate compositions of functions symbolically.
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The composition is evaluated explicitly when you supply a specific argument.
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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.
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Using Through, you can convert the expression to a more explicit form.
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This corresponds to the mathematical operator .
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Mathematica does not automatically apply the separate pieces of the operator to an expression.
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You can use Through to apply the operator.
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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.
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Functions like Expand do not automatically go inside heads of expressions.
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With the Heads option set to True, MapAll goes inside heads.
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The replacement operator /. does go inside heads of expressions.
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You can use Operate to apply a function specifically to the head of an expression.
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