For the functions f(x)=x^2 and g(x)=(x-7). Find if f(g(x)) = g(f(x)). What is g(f(2))?

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Since the composition of 2 functions is not commutative, then (fog)(x) is not the same with (gof)(x)

We'll get (fog)(x) for f(x) = x^2 and g(x) = (x-7).

(fog)(x) = f(g(x))

We'll substitute x by g(x) in the expression of f(x).

f(g(x)) = (g(x))^2

f(g(x)) = (x-7)^2 (1)

Let's determine (gof)(x) for f(x) = x^2 and g(x) = (x-7).

(gof)(x) = g(f(x))

We'll change x by f(x) in the expression of g(x).

g(f(x)) = (f(x)-7)

g(f(x)) = x^2 - 7 (2)

We notice that (1) is different of (2).

To calculate g (f (2)), we'll compute first f(2):

f(2) = 2^2 => f(2) = 4

g(f (2)) = g(4) <=> g(4) = (4-7) => g(4) = -3

g(f (2)) = -3

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