How to express a function as a function of another function?

[mnb96]

Hello,

I would like to know how I could approach the following problem. I am given two functions $y=f(x)$ and $z=g(x)$, and I would like to express the first function as a function of the second one: that is, $$y = h(z)$$, where h is not necessarily a linear function of z.

One explicit example could be: $$y=\frac{x}{a}$$ $$z=\frac{x}{a+b}$$

where the goal is to find a function h such that $$y=h(z)$$

 

[tiny-tim]

hello mnb96!

invert g (if you can) …

x = g^-1(z)

f(x) = f(g^-1(z)) ​

 

[mnb96]

Ups...:)

You are right. When the inverse for g exists, it is pretty easy. Thanks.
I was wondering if it is still possible to do something when an inverse does not exist, although this goes slightly beyond the original question.

 

[tiny-tim]

I was wondering if it is still possible to do something when an inverse does not exist, although this goes slightly beyond the original question.

it'd have to be a pretty weird function not to have at least a local inverse

 

[coelho]

it'd have to be a pretty weird function not to have at least a local inverse

Indeed... but while they may have a inverse, it may be hard (or even impossible) to write down this inverse function... unless "cheating" is allowed, like using functions like Maple's RootOf( f(x) ) ...

 

[tiny-tim]

Indeed... but while they may have a inverse, it may be hard (or even impossible) to write down this inverse function...

but we can still write it as g^-1