Hi,
First, determine domains of f and g, the domain of g is R, for f the domain is:
f = Sqrt[#^2 - 5 # - 6] &;
g = 4 # + 5 &;
Solve[f[x] == 0, x]
Plot[f[x], {x, -10, 15}]
wich gives (-inf,-1) and (6,+inf) .
Then for h=f(g):
h = ( # // g // f) &;
Solve[h[x] == 0, x]
Plot[h[x], {x, -10, 15}]
with a result (-inf,-3/2) and (1/4,+inf) .
The overall domain is (-inf,-3/2) and (6,+inf) .
Plot[{f[x], h[x]}, {x, -10, 15}]
For more complicated functions you may consider using NSolve or FindRoot.
I.M.
