discrete maths question

Question

Let f be the function {(1,4),(2,3),(3,1),(4,2)} and g be the function {(1,2),(2,3),(3,3),(4,3)}. For each of the functions h given below let c1=h(1), c2=h(2), c3=h(3) and c4=h(4). (So h is the function {(1,c1),(2,c2),(3,c3),(4,c4)}.)

if h = fof

h = fog

h = gof

what is c1c2c3c4 for each?

Answer 1

I'll show the case when h(x) = f o g(x) = f(g(x)).

c₁ = h(1) = f(g(1)) = f(2) = 3

c₂ = h(2) = f(g(2)) = f(3) = 1

c₃ = h(3) = f(g(3)) = f(3) = 1

c₄ = h(4) = f(g(4)) = f(3) = 1

The definition of g shows that g(1) = 2, g(2) = 3, g(3) = 3, and g(4) = 3. Those are the substitutions that I made above, and I then used the definition of f to determine f(g(x)).

Thus, when h(x) = f o g(x), h is the function {(1, 3), (2, 1), (3, 1), (4, 1)}.

Using this same procedure, you should be able to figure out the values of c for the other cases given.