<s>[INST] piecewise ceiling function and what $g(x) = f(1/x)$ means I have some limits I am supposed to find and I'm having trouble understanding the question. 
Given: 

$$f(x) = \begin{cases} 2 & \text{ if } \left \lceil x \right \rceil
\text{is even}\\-1   & \text{ if } \left \lceil x \right \rceil
\text{is odd} \end{cases}$$
$g(x) = f\left(\frac{1}{x}\right)$

what does $g(x) = f\left(\frac{1}{x}\right)$ mean? (maybe I'm making it more complicated than it should be in my head, it's supposed to be a 'challenge' problem.) 
example: $g\left(\frac{1}{3}\right)$, does that mean $g\left(\frac{1}{\frac{1}{3}}\right)$ so then $g(3) = -1$ ?
or like a composite function where $g\left(\frac{1}{3}\right)$ is odd so $f\left(\frac{1}{3}\right) = -1$ then $g(x) = \frac{1}{-1} = -1$?
I realize both come out to $-1$, but I think it makes a difference for when I have to find the limits. Or maybe not?
 [/INST] $g(x) = f(\frac 1x)$ means just what it looks like:
$g:x\to f(\frac 1x)$.
So for example:  If $x = \frac {\sqrt 3}4$ then 
$g(x)=g(\frac {\sqrt 3}4) = f(\frac 4{\sqrt 3}) = \begin{cases}2&\text{if }\lceil \frac 4{\sqrt 3} \rceil \text{ is even}\\-1&\text{if }\lceil \frac 4{\sqrt 3} \rceil \text{ is odd}\end{cases}$
$=2$
Or $g(52) = f(\frac 1{52}) = \begin{cases}2&\text{if }\lceil \frac 1{52} \rceil \text{ is even}\\-1&\text{if }\lceil \frac 1{52} \rceil \text{ is odd}\end{cases}$
$=-1$
......
Or we could simple say $g(x) =\begin{cases}2&\text{if }\lceil \frac 1{x} \rceil \text{ is even}\\-1&\text{if }\lceil \frac 1{x} \rceil \text{ is odd}\end{cases}$
.....
Might, or might not, be worth noting:
If $0< x \le 1$ then $f(x) = -1$ and if $-1 < x \le 0$ then $f(x) = 2$.
So if $x \ge 1$ then $0< \frac 1x \le 1$ so $g(x) = -1$.  And if $x \le -1$ then $-1 \le \frac 1x < 0$ so $g(x)=2$. 
And $g(0)$ is not defined.
</s>