g◦f(x) = g(f(x)) Note: Range of f must be in domain of g!

2) Suppose A = {1,2,3,4}, B={0,1,2}, C={1,2,3}. Let f : A→B be f= {(1,0),(2,1),(3,2),(4,0)} ,and g : B→C be g= {(0,1),(1,1),(2,3)} . Find g◦f.􏰛􏰜􏰛

g◦f = { (1,1), (2,1), (3,3), (4,1) }

4) Suppose A = {a,b,c}. Let f:A→A be the function f= {(a,c),(b,c),(c,c)}, and let g : A→A be the function g= {(a,a),(b,b),(c,a)}. Find g◦f and f◦g.

g◦f = { (a,a), (b,a), (c,a) } and f◦g={ (a,c), (b,c), (c,c) }

6) Consider the functions f,g:R→R defined as f(x)= 1/(x²+1) and g(x)=3x+2. Find the formulas for g◦f and f◦g.

g◦f (x) = 3(1/(x²+1))+2 f◦g (x) = 1/((3x+2)²+1)

8) Consider the functions f, g : Z×Z→Z×Z defined as f(m,n)=(3m−4n,2m+n) and g(m,n) = (5m+n,m). Find the formulas for g◦f and f◦g.

g◦f (m,n) = (5(3m−4n)+2m+n, 3m−4n) f◦g(m,n) = (3(5m+n)−4m,2(5m+n)+m)

10) Consider the function f :R² →R² defined by the formula f(x,y)= (xy,x³). Find a formula for f ◦ f .

f ◦ f (x,y) = (xyx³,(xy)³)