Pure Maths
 

Always we have given the COMPOSITE FUNCTIONS in that format i-e

f= something
g= something

and then we have to find "fog" and "gof" .....

but never in any book .. . composite functions are given then we have to find original functions...... i -e

fog= something
gof= something

f=??
g=??

...... if somebody have any idea about this problem then please discuss.....

(((AWA)))

One Approach
 

One approach can be this. We start with given: g(f(x))= something.

Now, we easily conclude: Dom(g(f(x))=Dom(f(x))
Range(g(f(x))=Range(g(y)) (assuming that g is defined over y)

Keeping this in mind we can arbitrarily make two functions out of the given g(f(x)) function so that all the variables are grouped in one function which we will call f(x) and the whole will be g(f(x)).

For example: Given: g(f(x))=3x^2-2x-9. (^ stands for power)
We can arbitrarily say, f(x)=3x^2-2x. So g(y)=y-9#

Now, if we are given some conditions regarding the functions f or g then we can't break g(f(x)) arbitratily. We have to break it in a way that the conditions for f and g are also satisfied.

I hope this helps you. If there are questions then I am interested too.

Composition functions
 

MR. ElementOf surprise!
u have written a thing which is mathematically incorrect.
i.e.
if g(f(x)) = something ...that [U]does not [/U] imply that dom (g (f(x)) = dom (f(x)).
Consider the following counter example.....
Let f(x) = x^2. then Dom (g(f(x)) = postive real numbers and zero...however dom(f(x)) = All real numbers.