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

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In summary, the conversation is about finding a function h such that y = h(z) when given two functions y=f(x) and z=g(x). The goal is to find an explicit example using y=\frac{x}{a} and z=\frac{x}{a+b}. It is also mentioned that if the inverse for g exists, it is easy to find the function h. However, when the inverse does not exist, it may be harder to write down h.
  • #1
mnb96
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Hello,

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

One explicit example could be: [tex]y=\frac{x}{a}[/tex] [tex]z=\frac{x}{a+b}[/tex]

where the goal is to find a function h such that [tex]y=h(z)[/tex]
 
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  • #2
hello mnb96! :smile:

invert g (if you can) …

x = g-1(z)

f(x) = f(g-1(z)) :wink:
 
  • #3
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.
 
  • #4
mnb96 said:
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 :wink:
 
  • #5
tiny-tim said:
it'd have to be a pretty weird function not to have at least a local inverse :wink:

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) ) ...
 
  • #6
coelho said:
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 :wink:
 

1. Can any function be expressed as a function of another function?

Yes, any function can be expressed as a function of another function. This is known as function composition, where one function is applied to the output of another function.

2. How do I express a function as a function of another function?

To express a function as a function of another function, you can use the composition notation f(g(x)), where g(x) is the inner function and f(x) is the outer function. This means that the output of g(x) is used as the input for f(x).

3. What is the purpose of expressing a function as a function of another function?

Expressing a function as a function of another function can help simplify complicated expressions and make them easier to understand and work with. It can also help in finding the inverse of a function.

4. Can you give an example of expressing a function as a function of another function?

One example is expressing the function f(x) = x^2 as a function of g(x) = x + 1. This can be written as f(g(x)) = (x + 1)^2. In this case, g(x) is the inner function and f(x) is the outer function.

5. Are there any guidelines or rules for expressing a function as a function of another function?

Yes, there are a few guidelines for expressing a function as a function of another function. These include making sure the inner function's output is a valid input for the outer function, and being mindful of the domain and range of the composed functions. It is also important to use the correct notation, with the inner function being enclosed in parentheses and the outer function written first.

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