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defining composite functions and inverses


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#1 SFNQuestions

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Posted 29 January 2017 - 02:01 AM

Suppose that for solving for the inverse of a function that f(x) can be manipulated into the form of g(f(h(x)))=x. Then, h(x) is inverted to show

g(f(x))=h^{-1}(x). Afterwards, the function is reverted to show h(g(f(x)))=x. Does this correctly show that h(g(x)) is in fact the inverse of f(x)? 

 

 


If not, what can be done in this situation to find the inverse of the original f(x)?


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#2 wtf

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Posted 29 January 2017 - 05:38 AM

Can you give an example?
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#3 SFNQuestions

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Posted 29 January 2017 - 06:35 AM

Can you give an example?

That's okay, I have done some more testing and found this method is valid. But, thank you for inquiring, it could have just have easily been the case that this was too complicated for me to determine. 


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#4 wtf

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Posted 29 January 2017 - 08:14 AM

Well I'm glad I could help.

Edited by wtf, 29 January 2017 - 08:15 AM.

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#5 Xerxes

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Posted 29 January 2017 - 06:41 PM

Suppose that for solving for the inverse of a function that f(x) can be manipulated into the form of g(f(h(x)))=x. Then, h(x) is inverted to show
g(f(x))=h^{-1}(x). Afterwards, the function is reverted to show h(g(f(x)))=x. Does this correctly show that h(g(x)) is in fact the inverse of f(x)?

No, it is not valid (or only trivially).

First some terminology.....

The domain of a function is the set of all those elements that the function acts upon. Each element in the set is called an argument for the function

The codomain - or range - of a function is the set of all elements that are the "output" of the function. So for any particular argument the element in the codomain is called the image of the argument under the function.

So, Rule 1 for function...No element in the domain may have multiple images in the codomain.

Rule 2 for functions.....Functions are composed Right-to-Left

Rule 3 for functions.......Functions can be composed if and only if the codomain of a function is the domain of the function that follows it (i.e. as written, is "on the Left)

Rule 4 for functions....... For some image in the codomain, the pre-image set is all those elements in the domain that "generate" this image. Notice that, although images are always single element, pre-images are sets - although they may be sets with a single member in which case the function is said to have an inverse - not otherwise.


Look closely at what you wrote above. and check how many of these rules are violated.
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#6 studiot

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Posted 29 January 2017 - 08:04 PM

Well said Xerxes. +1


SFNrtc, 

 

I recommend drawing diagrams to help understand Xerxes comments.


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#7 HallsofIvy

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Posted 17 February 2017 - 12:18 PM

Suppose that for solving for the inverse of a function that f(x) can be manipulated into the form of g(f(h(x)))=x. Then, h(x) is inverted to show

g(f(x))=h^{-1}(x). Afterwards, the function is reverted to show h(g(f(x)))=x. Does this correctly show that h(g(x)) is in fact the inverse of f(x)? 

 

 


If not, what can be done in this situation to find the inverse of the original f(x)?

 

  No, it doe not show that h(g) is the inverse of f.  You actually have to show two​ things- that h(g(f(x))= x and​ that f(h(g(x))= x.


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