Answer
$$f\left( t \right) = 10{e^{ - \frac{1}{2}t}}$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dt}} = - \frac{1}{2}y \cr
& {\text{Separate the variables}} \cr
& \frac{{dy}}{y} = - \frac{1}{2}dt \cr
& {\text{Integrate both sides}} \cr
& \int {\frac{{dy}}{y}} = - \int {\frac{1}{2}} dt \cr
& \ln \left| y \right| = - \frac{1}{2}t + C{\text{ }}\left( {\bf{1}} \right) \cr
& {\text{Using the initial condition }}\left( {0,10} \right) \cr
& \ln \left| {10} \right| = - \frac{1}{2}\left( 0 \right) + C \cr
& C = \ln \left( {10} \right) \cr
& {\text{Substitute }}C{\text{ into }}\left( {\bf{1}} \right) \cr
& \ln \left| y \right| = - \frac{1}{2}t + \ln \left( {10} \right) \cr
& {\text{Solve for }}y \cr
& y = 10{e^{ - \frac{1}{2}t}} \cr
& f\left( t \right) = 10{e^{ - \frac{1}{2}t}} \cr
& \cr
& {\text{Graph}} \cr} $$
