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About a function
Consider the following function:
$$ f: \mathbb{R} \to \mathbb{R}, \quad x \mapsto -\sin(1/x) $$
As I see it, the range of the function is $ \, ]-\infty,0[ \cup ]1,+\infty[ \, $ and in the range $ \, ]0,1[ \, $ it is only partially defined.
So the domain of the function is $ \, \mathbb{R} \setminus \{0\} \, $.
I want to calculate $ \, f^{ -1}(]-\infty,0[) \, $ and $ \, f^{ -1}(]-\infty,0[ \setminus \{1\}) \, $, but I am not sure what that means.
Can anybody tell me?
A:
Hint: If $y \in ]-\infty,0[$ then $f(y) = 0$. If $y \in ]0,1[$, then $f(y) = 0$, except at $y = 1$. If $y \in ]-\infty,0[ \cap ]0,1[$, then $f(y) \in ]-\infty,0[$, so that $f(y) \in \{0,-1,1\}$. Why is this?
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