Answer
$${f_{avg}} = \frac{{1 - {e^{ - 8}}}}{8}$$
Work Step by Step
$$\eqalign{
& {\text{The average value of the function is:}} \cr
& {f_{avg}} = \frac{1}{{4 - 0}}\int_0^4 {{e^{ - 2x}}} dx \cr
& {\text{Integrate and evaluate}} \cr
& {f_{avg}} = \frac{1}{4}\left[ { - \frac{1}{2}{e^{ - 2x}}} \right]_0^4 \cr
& {f_{avg}} = \frac{1}{4}\left[ { - \frac{1}{2}{e^{ - 2\left( 4 \right)}} + \frac{1}{2}{e^{ - 2\left( 0 \right)}}} \right] \cr
& {\text{Simplify}} \cr
& {f_{avg}} = \frac{1}{4}\left[ { - \frac{1}{2}{e^{ - 8}} + \frac{1}{2}} \right] \cr
& {f_{avg}} = \frac{1}{8} - \frac{1}{8}{e^{ - 8}} \cr
& {f_{avg}} = \frac{{1 - {e^{ - 8}}}}{8} \cr} $$
