when working with composite functions, does fog=gof? how do i create two functions,f(x) and g(x) to show that this statement is either true or false.i need to explain my reasoning
In mathematics, you cannot prove that a statement is true by a finite number of examples, because it is not possible to prove all possible cases by a finite number of examples.
However, to prove that a statement is false, you only need ONE counter-example. This is probably your case here.
Try
f(x)=x^2+2,
g(x)=x+2
then fog(x)=f(g(x))=(x+2)^2+2
gof(x)=x^2+4
Since we demonstrated fog(x)≠gof(x), the given statement is not true.
On the other hand, if we define
f(x)=2x, g(x)=4x
then fog(x)=f(4x)=8x^2
gof(x)=g(2x)=8x^2
so fog(x)=gof(x) in this case, BUT it does not prove anything.
To determine if the statement "fog = gof" is true or false, we need to look at the definition of composite functions.
The composite function (fog)(x) is created by first applying the function g(x) to x, and then taking the result and applying the function f(x) to it. Mathematically, it is represented as (fog)(x) = f(g(x)).
On the other hand, the composite function (gof)(x) is created by first applying the function f(x) to x, and then taking the result and applying the function g(x) to it. Mathematically, it is represented as (gof)(x) = g(f(x)).
So, in order to create two functions, f(x) and g(x), to show whether fog = gof, you need to carefully choose the functions and observe the results.
Let's consider an example:
Suppose f(x) = 2x and g(x) = x^2.
Now let's compute (fog)(x), (gof)(x), and compare the results.
1. (fog)(x) = f(g(x)) = f(x^2) = 2(x^2) = 2x^2
2. (gof)(x) = g(f(x)) = g(2x) = (2x)^2 = 4x^2
By comparing the results, we see that (fog)(x) = 2x^2 and (gof)(x) = 4x^2. Therefore, in this example, fog is not equal to gof.
Hence, by carefully choosing different functions for f(x) and g(x), you can show that the statement "fog = gof" may be true or false depending on the functions chosen. It is essential to evaluate the composition of functions by substituting values and simplifying them to determine if the statement is true or false.