Uncovering Unknown Functions: Solving for f(x) and g(x) in Complex Equations

[Kerbox]

Say you get a problem like this:
Find f(x) and g(x) when f(g(x))=|sin(x)| and g(f(x))=sin^2(sqrt(x)),
and Domain_f=R, Domain_g=[0,-> >

How would you approach to solve this, or do you have to keep guessing until you find two functions that fits?

 

[arildno]

Say you get a problem like this:
Find f(x) and g(x) when f(g(x))=|sin(x)| and g(f(x))=sin^2(sqrt(x)),
and Domain_f=R, Domain_g=[0,-> >

How would you approach to solve this, or do you have to keep guessing until you find two functions that fits?

You'll have to keep guessing until you see:
f(x)=sqrt(x)
g(x)=sin^2(x)
Are you sure about the domains?

 

[Kerbox]

I was looking for an algorithm or something that would work, when the example wasnt as simple as this one. When a lot of simplifying had been done to the expressions for example.

And of course, the domains are reversed. Sorry about that.

 

[arildno]

No, there are no general solve-all techniques for functional equations
(where your unknowns are functions, rather than some numbers, for example)

 

[geometer]

No, there are no general solve-all techniques for functional equations
(where your unknowns are functions, rather than some numbers, for example)

Of course mathematicians don't want to be perceived as just guessing at possible answers so they have termed this method "solution by inspection."

 

[arildno]

Of course mathematicians don't want to be perceived as just guessing at possible answers so they have termed this method "solution by inspection."

 

[Tide]

Hey, trial and error is a perfectly valid mathematical method! Of course it's not always the most efficient path to a solution. :-)

 

[arildno]

Hey, trial and error is a perfectly valid mathematical method! Of course it's not always the most efficient path to a solution. :-)

Well, it's a perfectly valid praxis, don't know about method though..