Help me for this question on composite functions
Does the composition of functions display the commutative property? Give an example of each case to illustrate your answer.
not usually, even for linear functions:
f = ax+b
g = cx+d
f◦g = ag+b = a(cx+d)+b = acx + ad+b
g◦f = cf+d = c(ax+b)+d = acx + bc+d
with suitable values for a,b,c,d it might work.
For other functions, all bets are off
f = e^x
g = x^2
f◦g = e^g = e^(x^2)
g◦f = f^2 = e^(2x)
No, the composition of functions does not display the commutative property. The composition of functions is not commutative because the order of composition matters.
To illustrate this, let's consider two functions:
Function f(x) = x^2, and function g(x) = 2x.
Now, let's evaluate the composition of these functions in two different orders:
Case 1: g(f(x))
g(f(x)) = g(x^2) = 2(x^2) = 2x^2
Case 2: f(g(x))
f(g(x)) = f(2x) = (2x)^2 = 4x^2
As you can see, the composition of functions g(f(x)) results in 2x^2, while the composition of functions f(g(x)) results in 4x^2. Since the outputs of the compositions differ depending on the order, we can conclude that composition of functions is not commutative.
To determine whether the composition of functions displays the commutative property, we first need to understand what the commutative property is.
The commutative property states that the order of the elements does not affect the result when performing an operation.
In the case of composition of functions, it means that the order in which we compose two functions should not affect the overall result.
To answer the question, we can examine two scenarios: one where the composition follows the commutative property, and one where it does not.
1. Commutative Property Example:
Let's consider two functions: f(x) = 2x and g(x) = x + 3.
Now, let's find the composition of f(g(x)) and g(f(x)):
- f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6
- g(f(x)) = g(2x) = 2x + 3
Since f(g(x)) = 2x + 6 and g(f(x)) = 2x + 3, we can see that f(g(x)) and g(f(x)) are not equal. Therefore, in this case, the composition of functions does not display the commutative property.
2. Non-Commutative Property Example:
Consider two different functions, h(x) = x^2 and k(x) = 2x.
Now, let's find the composition of h(k(x)) and k(h(x)):
- h(k(x)) = h(2x) = (2x)^2 = 4x^2
- k(h(x)) = k(x^2) = 2(x^2) = 2x^2
In this case, h(k(x)) = 4x^2 and k(h(x)) = 2x^2, which are not equal. Therefore, the composition of functions does not display the commutative property.
In conclusion, the composition of functions does not generally exhibit the commutative property. The order in which functions are composed can affect the final result.