Why Can We Swap Variables When Finding Inverse Functions?

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To find the inverse of a function y=f(x), the process involves solving for x in terms of y, resulting in x=g(y). This leads to the equation y=g(x) after swapping x and y. The relationship between g(f(x))=x and g(y)=x highlights that both expressions represent the same inverse function concept. The ability to swap variables stems from the definition of inverse functions, where the output of one function becomes the input of the other. Understanding this relationship clarifies the process of finding inverse functions.
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We have y=f(x), and get the inverse by uing the first function and solving it for x and get x=g(y). (F and g are different functions.) Then we swap the name of x and y and we get y=g(x).

Buw why can we do this when we want to find the inverse functions? If we got y=f(x) and want to find the inverse we take g(f(x))=x. But how is this related to the first thing I did?
 
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Since y= f(x), g(f(x))= x is the same as g(y)= x.
 

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