finding domain of (g) and (f)

Question

suppose f(x) = {(1,10)(2,20)(3,30)(4,40)} and g(x) = {(10,3)(20,1)(7,2)(3,40)}

 

Find dom(g o f)

 

 

and find dom(f o g)

Answer 1

(g o f)(x) = g(f(x))

 

 dom(g o f) = {x l x ∈ domf and f(x) ∈ domg} = {1, 2}

 

   Note:  1 is in dom f and g(f(1)) = g(10) = 3

             2 is in dom f and g(f(2)) = g(20) = 1

             3 is in domf, but g(f(3)) = g(30), which is not defined

             4 is in domf, but g(f(4)) = g(40), which is not defined

             So, 1 and 2 are in dom(g o f), but 3 and 4 are not. 

 

(f o g)(x) = f(g(x))

 

  dom(f o g) = {x l x ∈ domg and g(x) ∈ dom f} = {7, 10, 20}

 

   Note:  7 is in domg and f(g(7)) = f(2) = 20

            10 is in domg and f(g(10) = f(3) = 30

            20 is in domg and f(g(20) = f(1) = 10

             3 is in domg, but, f(g(3)) = f(40), which is undefined.

            So, 7, 10, and 20 are in dom(f o g), but 3 is not.