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Find the range of f(x)=1-|x-2|

Given, fx=1-x-2Now consition will be applied on x-2 not on 1-x-2.So  x-2=x-2, when x>2-x+2 , when x<2So we get fx=1-x-2, when x>21--x-2 , when x<2=1-x+2, when x>21+x-2 , when x<2=3-x, when x>2-1+x, when x<2Now max value will be obtained when in 1-x-2 , x-2=0 because 1-something will always be negative for higher values of xSo max value = 1-0 = 1and minimum value will be -∞, because domain of function f is x∈R

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Experts when I solve by defining |x-2| I do like
case 1-x-2>or=0
1-x+2=y
x=3-y
So y belongs to R
Case 2- x-2<0
1+x-2=y
x=y+1
So y belongs to R again. Their union is again R.
But this answer is wrong. How is it wrong. Which step have I done wrong? Ans is (-infinity,1] Thanks!
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