• Posted: Jan 16, 2023
• By:

Surah Yateen zilal-Alam Namoos 23 -.

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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