Interpret this function notation? I have the following function notation
\begin{align} 
f: &\, \mathbb{R} \rightarrow \mathbb R^2 \\ &x\mapsto y=f(x) 
\end{align}
Does it actually mean 
\begin{gather} 
y=f(x) \\ 
(y_1,y_2)=(f_1(x),f_2(x)) \\ 
\begin{cases}
y_1=f_1(x) \\ 
y_2=f_2(x)\end{cases} \qquad ? 
\end{gather}
Or with an exemple, if $f(t)=(e^t,t^2)$ and $y=f(t)$, so 
\begin{align}  
(y_1,y_2)=(f_1(t),f_2(t))=(e^t,t^2)
\end{align}
\begin{cases} 
y_1=f_1(t)=e^t\\ y_2=f_2(t)=t^2
\end{cases}
Is this correct?

Yes. The notation $f:\mathbb{R}\to\mathbb{R}^2$ tells you that the function takes in a single real number and returns an ordered pair of real numbers. Hence the notation $x\mapsto y=f(x)$ is telling you that $y$ is an ordered pair of real numbers.
