I need the domains of f(g(x)), g(f(x))... etc. Help?

Question

f(x) = x+(1/x)

g(x) = (x+1)/(x+2)

I need the domains of f(g(x)), g(f(x)), f(f(x)) and g(g(x)).

Answer 1

domain of f(x): x <> 0 or (-inf , 0) U (0 , inf)

domain of g(x): x <> -2 or (-inf , -2) U (-2 , inf)

remember for composites, the first part of the domain is the domain of the "inner function" (g(x) for the next problem, which is where x <> -2 comes from)

domain of f(g(x)) : g(x) <> 0 ==> x <> -1 and x <> -2

interval notation: (-inf , -2) U (-2,-1) U (-1 , inf)

domain of g(f(x)): f(x) <> -2 ==> x + 1/x <> -2

x^2 + 1 + 2x <> 0 ==> (x + 1)^2 <> 0 ==> x <> -1

therefore domain is x <> -1 and x <> 0

interval notation: (-inf, -1) U (-1,0) U (0, inf)

domain of f(f(x)) : f(x) <> 0 ==> x + 1/x <> 0 ==> x^2 + 1 <> 0 (all real numbers); therefore domain is x <> 0 (to ensure that the inner f(x) is defined)

interval notation: (-inf , 0) U (0 , inf)

domain of g(g(x)): g(x) <> -2 ==> -2(x + 2) <> x + 1

-2x - 4 <> x + 1

3x <> -5 ==> x <> -5/3

domain of g(g(x)) is x <> -5/3 and x <> -2

interval notation: (-inf , -2) U (-2 , -5/3) U (-5/3, inf)