Can someone help me ? I'm really confused with these questions I do not know how to get the inverse ! and I'm not sure how to prove it's one to one
Suppose $f: \mathbb{R} \to \mathbb{Z}$ where $f(x) = \lceil 2x - 1 \rceil$.(ceiling function)
i. Is $f$ one-to-one? Explain.
ii. If $A = \{x \mid 1 \leq x \leq 4\}$, find $f(A)$.
iii. If $B = \{3, 4, 5, 6, 7\}$, find $f(B)$.
iv. If $C = \{-9, -8\}$, find $f^{-1}(C)$.
v. If $D = \{0.4, 0.5, 0.6\}$, find $f^{-1}(D)$.
Hint:
Of course you can't get the inverse of $f$: $f(x)=\lceil 2x-1\rceil$ does not define a one-to-one function. Actually, $f(x)=n$ means that $$n-1< 2x-1\le n\iff n<2x\le n+1\iff \frac n2<x\le \frac n2+\frac12,$$ so it's locally constant (and non-decreasing).
On the other hand, in the last question $f^{-1}(C)$ does not denote the image of $C$ by the inverse function of $f$, which does not exist, but the inverse image of $C$ by the function $f$, i.e. $$f^{-1}(C)=\{x\in\mathbf R\mid f(x)\in C\,\},$$ and this one always exists (possibly empty).