# Given f(x) and g(x), please find (fog)(X) and (gof)(x) f(x) = 2x   g(x) = x+3

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f(x) = 2x

g(x) = x + 3

First let us find (fog)(x)

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

= f(x+3)

= 2(x+3)

= 2x + 6

==> (fog)(x) = 2x + 6

Now let us find (gof)(x):

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

= g(2x)

= 2x + 3

==> (gof)(x) = 2x + 3

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So remember that `(f@g)(x)=f(g(x))`

so subbing in g(x), we get

`f(x+3)`

Replacing each x in f with x+3, we get

`2(x+3)`

`(f@g)(x)=2x+6`

``

For `(g@f)(x)=g(f(x))`

We have that `(g@f)(x)=g(f(x))`

`=g(2x)`

`(g@f)(x)=2x+3`

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To calculate the compositions of the functions, we'll apply the rule:

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

It is obvious that we'll substitute x by the expression of g(x) and we'll get:

f(g(x)) = f(x+3) = 2*(x+3)

We'll remove the brackets and we'll have:

(fog)(x) = 2x + 6

Now, we'll calculate  (gof)(x).

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

It is obvious that we'll substitute x by the expression of f(x) and we'll get:

g(f(x)) = g(2x) = 2x + 3

(gof)(x) = 2x + 3

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f(x) = 2x. g(x) = x+3.

To find (fog)(x) and (gof)(x).

Solution:

f(x) = 2x. g(x) = x+3.

Substitute g(x) for x in f(x) to get (fOg)(x):

f((g(x)) = 2 (g(x) = 2(x+3) = 2x+6.

To get (gof)(x), substitute  f(x) for x in g(x):

(gof)(x) =  (2x) + 3 = 2x+3.