calculus

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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

  • calculus -

    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.

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