Cases inside tabular environment with left aligned column

Question

I want to have equations in a column of a table, using the amsmath cases option, but where the column is left aligned. The MWE below shows an example of the equations that I need in the column, with the second table showing the layout I am after.

What I found online suggests you need the p{} option in order to include the cases option from amsmath. The array package has been suggested as a way to get the left aligned in the second column, but I cannot get it working as I expected it to.

Appreciate any pointers to what I am doing wrong.

\documentclass[a4paper,10pt]{article}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{array}

\begin{document}
%
\begin{table}
 \begin{tabular}{| c>{\raggedright\arraybackslash}p{10cm} |} \hline
Option 1 & \[M_f(t) =  \begin{cases} 
      M_f(0) & t \leq \tau \\
      M_f^\dag & t > \tau 
   \end{cases} \]\\ \hline
Option 2 & \[ M_f(t) = \begin{cases}
            \left[M_f^\dag - M_f(0)\right]\left[\frac{\exp(-0.25 t) - 1}{\exp(-0.25 \tau) - 1}\right] + M_f(0) & t \leq \tau  \\
            M_f^\dag & t> \tau
                \end{cases} \]  \\ \hline
 \end{tabular}
 \caption{The equations I need}
 \label{a label}
\end{table}


\begin{table}
 \begin{tabular}{| l  l |} \hline
Option 1 & $x^2 + 3$ \\ \hline
Option 2 & $\sin(x) + \text{a much longer equation than the others to show the formatting}$ \\ \hline
 \end{tabular}
 \caption{The layout I want}
 \label{a 2nd label}
\end{table}

\end{document}

Answer 1

like this?

\documentclass[a4paper,10pt]{article}
\usepackage{amsmath, amssymb}
\usepackage{makecell}

\begin{document}
\begin{table}
    \centering
    \setcellgapes{3pt}
    \makegapedcells
\begin{tabular}{| c l|}
    \hline
Option 1 &  $ M_f(t) =  \begin{cases}
                        M_f(0)      & t \leq \tau \\
                        M_f^\dag    & t > \tau
                        \end{cases}$    \\
    \hline
Option 2 &  $ M_f(t) = \begin{cases}
                        \left[M_f^\dag - M_f(0)\right]
                        \left[\dfrac{\exp(-0.25 t) - 1}
                                    {\exp(-0.25 \tau) - 1}\right]
                        + M_f(0)    & t \leq \tau  \\
                        M_f^\dag    & t> \tau
                        \end{cases}$    \\
    \hline
\end{tabular}
\caption{The equations I need}
\label{a label}
\end{table}
\end{document}