Identity map and Inverse Image

[wayneckm]

Hello everyone,


I would like to ask what's the purpose of identity map? Recently I came across something that apparently use this to find the inverse image of a function $$F(x)$$ in the form of $$F(x) = ( f(x) , x )$$.

Thanks.


Wayne

 

[HallsofIvy]

The "identity map" is just very simple function that maps everything to itself, f(x)= x. In the operation of "composition" of functions, it asks the same way the number "0" does with addition or the number "1" does with multiplication.

In particular, just as the "negative" (additive inverse) is such that (-x)+ x= 0 and the "reciprocal" (multiplicative inverse) is such that (1/x)*(x)= 1, so the "inverse function" is defined as the function, $f^{-1}(x)$ such that $f(f^{-1}(x)= x$ and $f^{-1}(f(x))= x$- both compositions giving the identity function.

Another way of looking at "inverse" functions is that they "reverse" the original function. If y= f(x), then $x= f^{-1}(y)$: if f "changes" x into y, then $f^{-1}$ changes y into x.

In particular, if we write a function, f. as a set of "ordered pairs", {(x, y)} where $y= f(x)$, then its inverse function reverses those pairs- it is {(y, x)}= {f(x), x}.

The "purpose" of the identity map is really to allow us to define the "inverse" function and that allows us to solve equations: If we know that f has inverse $f^{-1}$ (not all functions have inverses) then we can solve f(x)= a by taking $f^{-1}$ of each side: $f^{-1}(f(x))= f^{-1}(a)$ and, since taking a function and then its inverse gives the "identity map", we have $f^{-1}(f(x))= x$ so that equation becomes $x= f^{-1}(a)$.