Question

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

Answer 1

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.