Piecewise function

f(f(x)) = f(b)

f(b) = 4    for the first condition:    x-3 = 4    x=7   this is not <=0   so it does not apply

                for the second condition    5x+2 =4    x = 2/5    which IS greater than 0

                       so b = 2/5

Now do this again

f( x)  = 2/5     for the first condition   x-3 = 2/5    x = 3 2/5     this does not apply because x > 0

                        so the second condition applies 

                               5x+2 = 2/5

                                 x =  -8/5 / 5 = -8/25     this does not apply either

 

There is no value for  which this is true

 

Here is a graph of the piecewise function.....you will see no value where it = 2/5   ( the green line does not intersect the graph)

https://www.desmos.com/calculator/qt7spyez1o