How to find the domain of composition of 2 functions

Questions
1

All:

How to find the domain of composition of 2 functions :
f(g(x) where f(x)= radical( x^2 -5x -6) and g(x) = 4x + 5
using Mathematica

Thanks for your help

Regards,
Bzoo

2

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.