JEE Math Practice Question

If $$f\left( y \right) = {e^y},$$ $$g\left( y \right) = y;y > 0$$ and $$F\left( t \right) = \int\limits_0^t {f\left( {t - y} \right)g\left( y \right)dy,} $$ then :

1. A.$$F\left( t \right) = t{e^{ - t}}$$
2. B.$$F\left( t \right) = 1t - t{e^{ - 1}}\left( {1 + t} \right)$$
3. C.$$F\left( t \right) = {e^t} - \left( {1 + t} \right)$$
4. D.$$F\left( t \right) = t{e^t}$$.