Inverse function for several variables

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SUMMARY

The discussion centers on the concept of finding an inverse function for a multivariable function, specifically examining the function f(x,y,z) = xyz. It concludes that this function does not possess an inverse g(x,y,z) because it is not one-to-one; multiple combinations of (a, b, c) yield the same product d. The inability to establish a unique mapping from the output back to the input is highlighted as a critical factor in the absence of an inverse.

PREREQUISITES
  • Understanding of multivariable functions
  • Knowledge of one-to-one and onto functions
  • Familiarity with inverse functions in mathematics
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of one-to-one and onto functions in detail
  • Explore the concept of inverse functions for multivariable cases
  • Investigate examples of functions that do have inverses
  • Learn about the implications of non-injective functions in calculus
USEFUL FOR

Mathematicians, students studying advanced calculus, and anyone interested in the properties of functions and their inverses.

zetafunction
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in one dimension one have that for a function [tex]f(x)[/tex] we can define another function [tex]g(x)[/tex] so [tex]f(g(x)=x[/tex]

my question or problem is the following, if i have a function of three variables [tex]f(x,y,z)[/tex] then i can define another function [tex]g(x,y,z)[/tex] so [tex]f(g(x,y,z))=Id[/tex]

for example for the function [tex]f(x,y,z)= xyz[/tex] what would be its inverse g??
 
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zetafunction said:
in one dimension one have that for a function [tex]f(x)[/tex] we can define another function [tex]g(x)[/tex] so [tex]f(g(x)=x[/tex]

my question or problem is the following, if i have a function of three variables [tex]f(x,y,z)[/tex] then i can define another function [tex]g(x,y,z)[/tex] so [tex]f(g(x,y,z))=Id[/tex]

for example for the function [tex]f(x,y,z)= xyz[/tex] what would be its inverse g??

It doesn't have an inverse because it is not 1-1 onto any point any point in its range. For example if d = abc then d also equals bac, cba etc.
 

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