Answer:
The correct option are;
f(x) = x² - 4 and g(x) = x - 3, (f ο g)(2) = -3 and (g ο f)(2) = -3, so the function is commutative
Given f(x) = 4·x and g(x) = x², (f ο g)(x) = 4·x² and (g ο f)(x) = 16·x²
So the function is not commutative
Step-by-step explanation:
For the equations f(x) = x² - 4 and g(x) = x - 3, we have;
(f ο g)(x) = f(g(x)) = (x - 3)² - 4 = x² - 6·x + 9 - 4 = x² - 6·x + 5
At x = 2, we have;
(f ο g)(2) = f(g(2)) = 2² - 6×2 + 5 = - 3
Similarly, we have;
(g ο f)(x) = g(f(x)) = x² - 4 -3 = x² - 7
At x = 2, we have;
(g ο f)(2) = g(f(2)) = 2² - 7 = 4 - 7 = -3
Therefore, by commutative property, we have that the result of an operation does not change by changing the order of the operands such that we have;
a + b = b + a or a·b = b·a from which we have resolved also the following operation is commutative
(f ο g)(2) = (g ο f)(2)
Similarly given f(x) = 4·x and g(x) = x², (f ο g)(x) = 4·x² and (g ο f)(x) = 16·x²
So the function is not commutative
(f ο g)(x) = f(g(x)) = 4·x²
(f ο g)(x) = 4·x²
(g ο f)(x) = g(f(x)) = (4·x)² = 16·x²
(g ο f)(x) = 16·x²
∴ (f ο g)(x) = 4·x² ≠ (g ο f)(x) = 16·x²
(f ο g)(x) ≠ (g ο f)(x) the function is not commutative.
