Composite Function.





What is Composite Function?

Let f:A→B and g:B→C be any two functions , we can represent this two functions in a diagram as following.

Function "g" of another function"f".

What is happening here is the output of function “f” is again processed by another function “g” to give a new output.

If we make a new function (let the new function be “h”) which maps input of the function “f” directly to output of function “g” or maps “A” directly to “C” then the new function “h” is said to be the composite function of functions “f” and “g”.

Thus,

A single function formed by the combination of two or more function is said to be a composite function.

NOTE:

Composite function of “f” and “g” is not same as the composite function of “g” and “h”.

Denotation of Composite Function:

We denote a composite function of two functions (let “f” and “g”) by adding a “o” sign in between them , and the two functions “f” and “g” are ordered on the basis of which occurs first.

Like we denote composite function of f:A→B and g:B→C as:

gof

NOTE:The function which occurs at first is placed in second place of composite function symbol and vice versa.

The composite function of ¬†Functions “f” and “g” in above diagram or “gof” is shown diagrammatically as following:

Composite Function of Functions "f" and "g".

Some Examples of Composite Function.

a> If f:A→B and g:B→C are two functions defined by:

Functions "f" and "g".

Then , (gof):A→C or composite function of “f” and “g” is defined by:

Composite Function of "f" and "g".

b> If f:R→R is defiend by f(x)=3x and g:R→R is defined by g(x)=x+1 and “R” is the set of real numbers.

Then ,

(gof) is defined by:

(gof)(x)=g(f(x))

=g(3x)

=3x+1

In above calculation , in g(f(x)) first function “f” acts on “x” and changes it to “3x” as “f’ is defined and changes g(f(x)) into g(3x) and then , function “g” acts on “3x” to change it to “3x+1″ is function “g” is defined.



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