Finding the domain and the range of $\cot x$

Question

I need to find the domain and the range of this function:

$$h(T) = \cot T$$

Please help me.

Answer 1

Domain: $\mathbb R-{k\pi} ,k \in \mathbb Z$
Range : $\mathbb R$
if $f(x)=cot(x)$ is $\frac{\cos(x)}{\sin(x)}$ , domain of a function is set of the points where $f(x)$ is defined. In a fraction, the denominator can't be zero. So:
$\sin(x) \neq 0 \implies x \neq k\pi ,k \in Z= \{\dots,-2,-1,0,1,2,\dots\}$

Range of a function is set of the points $f(x)$ can generate.
In this case, $\cot(x)$ can generate all the numbers in $\mathbb R$ (real numbers).