Expert: Paul Klarreich - 2/9/2007

Question
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=??


any idea then waiting for ur answer???????????  

Answer
Questioner:   Abdul Waqas
Category:  Calculus << PRECALCULUS
 
Subject:  Composite Functions
Question:  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=??


any idea then waiting for ur answer???????????
..........................................
Hi, Abdul,

When you put two functions together as a composite, then you get a new function.  But you want to take a function which (you believe) was produced as a composition and derive the originals.

For example, if  

f(x) = 3x
g(x) = x^2 + 1

then (I am going to use the notation  f(g(x)) instead of your  f o g.)

g(f(x)) = (f(x))^2 + 1 = (3x)^2 + 1 = 9x^2 + 1

Are you asking for a way to take the result,  9x^2 + 1, and derive the original two functions f,g?

THERE IS NO UNIQUE WAY TO DO THIS.  There will always (well, almost always) be more than one way to 'decompose' the resulting function.

For example:

f(x) = x^2
g(x) = 9x + 1

You can easily check that THIS pair of functions gives:

g(f(x)) = 9x^2 + 1,  same as before.

And how about:

f(x) = 9x^2
g(x) = x + 1

And I am sure you will be able to come up with other possibilities.  Now you know why your textbook does not have any problems like this -- the 'answers in the back of the book' section would be bigger than the rest of the book.