If f is a function which maps x to y, then f^(-1) is a function that maps y to x. However, if f(x) maps more than one x value to a single y, then there is no way to know which x f^(-1) should map y to.
If you don't care about about the inverse being a function, then we can just consider all the cases, similar to how we invert y=x^2 by swapping y and x and then solving for y: x = y^2 => y = +-sqrt(x). Doing so for f(x) = |2-x|, we get
x = |2-y|
=> x = +-(2-y)
=> +-x = 2-y
=> y = 2+-x
So the inverse is y = 2+-x. Note that as f(x)>=0 in the initial function, we must have x>=0 for the inverse. This gives us the graph
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which is a reflection of y = |2-x| across the line y=x, as we would expect .
If we want to have the inverse as a function, then we must consider a restriction of |x-2| to a domain in which every x value is mapped to a distinct y value (a function with this property is called one-to-one).
Again, this is analogous to finding the inverse of g(x)=x^2. If we add the initial restriction x <= 0, then we get x = -sqrt(x^2) => g^(-1)(x) = -sqrt(x). If we add the initial restriction x>=0, then we get x = sqrt(x^2) => g^(-1)(x) = sqrt(x). Notice that we get a different inverse function depending on how we restrict the domain of g(x).
Because f(x) is defined as
f(x) = {(-(x-2) if x-2 <= 0), (x-2 if x-2>=0):}
=> f(x) = {(2-x if x <= 2), (x-2 if x>=2):}
natural restrictions would be considering f(x) = |2-x|, x<=2 or f(x) = |2-x|, x>=2. Then we are just finding the inverse of 2-x or x-2 on the restricted interval.
