Determining the inverse image of $\{1,2,5\}$ under a ceiling function

Question

I'm trying to find the inverse image of a set under a ceiling function. Specifically, if $g: (0,\infty) \rightarrow \mathbb{N}$ is the function $g(x) = \lceil x \rceil$, what is the inverse image of the set $A =\{1,2,5\}$ under $g.$

From the definition of an inverse image, I believe that $$g^{-1}[A] = \{x\in (0,\infty)\mid g(x) \in A\},$$ but after this I am very lost as to how to proceed; may I ask for some help?

Answer 1

For $1$, we need to find all possible $x$, such that $g(x)=1$. Let's see some points to illustrate this. For example,

if $x=0,$ then $g(0)=0$

if $x=0.1,$ then $g(0.1)=1$

if $x=0.9,$ then $g(0.9)=1$

if $x=1,$ then $g(1)=1$

if $x=1.1,$ then $g(1.1)=2$

So we can see the inverse image for $1$ is

$$g^{-1}(1)=(0,1]$$

Can you proceed to find the inverse image for $2$ and $5$?