A detail method.
Let us write g(g(g(g(g(x))))) as g5(x) and g(g(g(g(x)))) as g4(x) and so on.
Or g5(x) = 19.
Note that if x is odd, g(x) = x + 5 and hence g(x) is always even for x as odd.
So if g(x) is odd, x has to be even.Or if g5(x) is odd, and since g5(x)) = g(g4(x)), g4(x) is even.
Hence, if g5(x) = 19, g(g4(x)) = 19 = [g4(x)]/2.
Or g4(x) = 38.
Or g(g3(x)) = 38.
Now, g3(x) can be both odd or even.
If g3(x) is odd, g(g3(x)) = 38 = g3(x) + 5.
Or g3(x) = 33.
Now, since g3(x) is odd and g3(x) = g(g2(x)), g2(x) is even.
Since g(g2(x)) = 33 = [g2(x)]/2.
Or g2(x) = 66.
Or g(g(x)) = 66.
If g(x) is odd, 66 = g(x) + 5. Or g(x) = 61 if g(x) is odd.
Since here 61 is odd, x has to be even.
Or g(x) = 61 = x/2.
Or x = 122 if g(x) is odd. x1 = 122
If g(x) is even g(g(x)) = 66 = g(x)/2 .
Or g(x) = 132.
Now if x is odd, g(x) = 132 = x+5.
Or x = 127, if x is odd. x2 = 127.
If x is even, g(x) = 132 = x/2.
Or x = 264, if x is even. x3 = 264.
We can summarize and say that if g3(x) is odd, there are 3 values of x.
Let us see what happens if g3(x) is even.
So, then g(g3(x)) = 38 = [g3(x)]/2.
Or g3(x) = 76.
So g(g2(x)) = 76.
If g2(x) is odd, 76 = g2(x) + 5.
Or g2(x) = 71.
Or g(g(x)) = 71.
Again, since 71 is odd, g(x) has to be even.
Or 71 = g(x)/2.
Or g(x) = 142.
If x is odd = 142 = x + 5.
Or x = 137, So x4 = 137.
If x is even, 142 = x/2.
Or x = 284. Or x5 = 284.
So we get 2 more values of x if g2(x) is odd.
If g2(x) is even 76 = g2(x)/2.
Or g2(x) = 152.
So g(g(x)) = 152.
If g(x) is odd, 152 = g(x) + 5.
Or g(x) = 147.
Since 147 is odd, x is even . Or 147 = x/2.
Or x = 294, x6 = 294.
If g(x) is even, 152 = g(x)/2.
Or g(x) = 304.
So if x is odd, 304 = x + 5.
Or x = 299, x7 = 299.
If x is even, 304 = x/2.
Or x = 608. x8 = 608.
Or there are 8 value of x.
