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Finding range of a function using inequalities

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Homework Statement
If ##f : R \Rightarrow R##and ##g:R \Rightarrow R## defined by ##f(x) = |x|## and ##g(x)=(x-3)## then ##{g(f(x)) : -1.6 < x < 1.6}##
Homework Equations
Given ##f(x)## and ##g(x)## for ## -1.6 < x < 1.6## we get ##0\leq f(x)<1.6##
Now for ##0\leq f(x)< 1## ##\implies## ##g(f(x))=-3## since ## -3\leq f(x)-3<-2##
Again for, ##1\leq f(x)< 1.6## ##\implies## ##g(f(x))=-2## since ## -2 \leq f(x)-3 < -1.4##
Thus the required set is {-3,-2}
My attempt :

Given ##f(x)## and ##g(x)## for ## -1.6 < x < 1.6## we get ##0\leq f(x)<1.6##
Thus, for ##f(g(x))## we get ## -3 \leq g(f(x)) < -1.4##
Thus the required set should be the interval ##[-3, -1.4)##?

My Questions :
1. What have I missed since my answer does not match the given solution.
2. In the given solution why is the ##f(x)## inequality broken into ##0\leq f(x)< 1## and ##1\leq f(x)< 1.6##
3. After splitting the inequality I don't understand how the result is not an interval but two unique integers.
 
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PeroK

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Your answer is correct given the problem stated. The other answer may be the answer to a different question.
 

WWGD

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My attempt :

Given ##f(x)## and ##g(x)## for ## -1.6 < x < 1.6## we get ##0\leq f(x)<1.6##
Thus, for ##f(g(x))## we get ## -3 \leq g(f(x)) < -1.4##
Thus the required set should be the interval ##[-3, -1.4)##?

My Questions :
1. What have I missed since my answer does not match the given solution.
2. In the given solution why is the ##f(x)## inequality broken into ##0\leq f(x)< 1## and ##1\leq f(x)< 1.6##
3. After splitting the inequality I don't understand how the result is not an interval but two unique integers.
You may want to use \mathbb r wrapped around in Latex for the Real numbers, instead of just R. It comes out like this ##\mathbb R##. In some cases you may eliminate ambiguity.
 

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