Interpret this notation (ODE) I need help how to interpret the notation in the following IVP:

\begin{align}
\dot x&=f(t,x), \quad \tag 1\\
x(t_o)&=x_0, \quad \tag 2 
\end{align}
  We assume $f\in C(U,\mathbb R^n)$, where $U$ is an open subset of $\mathbb R^{n+1}$ and $(t_0,x_0)\in U$.

Q1: Does it mean I actually have the following:
Vector-valued functions:
\begin{align}
x&:\mathbb R \rightarrow \mathbb R^n, \quad x(t)=(x_1(t), x_2(t), \dots, x_n(t))
\tag 3
\\
\dot x&:\mathbb R \rightarrow \mathbb R^n, \quad  
\dot x(t)=(\dot x_1(t), \dot x_2(t), \dots, \dot x_n(t))
\tag 4
\\
f&:\mathbb R^{n+1} \rightarrow \mathbb R^n \tag 5
\end{align}
We can write $f$ more precisely: $f:U \rightarrow \mathbb R^n$, $U\subset \mathbb R^{n+1}$.
Vector (constant vector):
\begin{align}
x_0\in \mathbb R^n, \quad x_0=(x_{0_1}, x_{0_2},\dots ,x_{0_n}) \tag 6 
\end{align}
Scalars:
\begin{align}
t\in \mathbb R\\
t_0\in \mathbb R
\end{align}
Q2:
Is the explicit form of $f$ the following:
\begin{align}
f(t,x_1(t), x_2(t), \dots, x_n(t))=
\large(
&f_1(t,x_1(t), x_2(t), \dots, x_n(t)),\\
&f_2(t,x_1(t), x_2(t), \dots, x_n(t)),\\
&\qquad \qquad \qquad  \vdots \\
&f_n(t,x_1(t), x_2(t), \dots, x_n(t))\large)
\end{align}

Your interpretation is the way most people I know would interpret that, and I can't see how else you could interpret it. 
