Suppose A = {1, 2, 3}. Let f : A → A be the function f = {(1, 2), (2, 2), (3, 1)}, and letg : A → A be the function g = {(1, 3), (2, 1), (3, 2)}. Find g ◦ f and f ◦ g

To find g ◦ f, we need to perform the composition of functions g and f.

Step 1: Start with the function g and substitute the outputs of f as inputs into g.

g ◦ f = g(f(1)), g(f(2)), g(f(3))

Step 2: Evaluate the composition for each input in A.

For f(1), we have f(1) = 2. So, g(f(1)) = g(2) = 1.

For f(2), we have f(2) = 2. So, g(f(2)) = g(2) = 1.

For f(3), we have f(3) = 1. So, g(f(3)) = g(1) = 3.

Therefore, g ◦ f = {(1, 1), (2, 1), (3, 3)}.

To find f ◦ g, we need to perform the composition of functions f and g.

Step 1: Start with the function f and substitute the outputs of g as inputs into f.

f ◦ g = f(g(1)), f(g(2)), f(g(3))

Step 2: Evaluate the composition for each input in A.

For g(1), we have g(1) = 3. So, f(g(1)) = f(3) = 1.

For g(2), we have g(2) = 1. So, f(g(2)) = f(1) = 2.

For g(3), we have g(3) = 2. So, f(g(3)) = f(2) = 2.

Therefore, f ◦ g = {(1, 1), (2, 2), (3, 2)}.