Here f(x) and g(x) is defined .We are asked to compute the composition functions (fog)(x) and (gof)(x).

Now (fog)(x)

=f(g(x))

=f(x+3)

=2(x+3)

=2x+6

Also (gof)(x)

=g(f(x))

=g(2x)

=2x+3

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I like to write composition problems such as this one as g(f(x)) and f(g(x)) rather than the (fog)(x) and (gof)(x). I do this because it reminds me the order in which to solve the problem. When we are finding g(f(x)), I want to work inside out. This means since f(x) is the inner most part of the equation, I want to take f(x) and plug it into g(x), as shown below:

g(f(x)) = g(2x) <---- I replaced the f(x) with 2x because they are equal

In the g(x) equation, we need to replace the x with 2x, so:

g(2x) = 2x + 3

There is nothing to simplify in the above equation, so we now know that g(f(x)) = 2x + 3

The process is similar for f(g(x)).

f(g(x)) = f(x+3) <------ Again, working inside out.....g(x) is replaced with x + 3 because they are equal.

f(x +3) = 2(x + 3) <------x in the f(x) equation is replaced with x + 3

= 2x + 6 < ----- distributing the 2 to the x and the 3 gets the final answer

so f(g(x)) = 2x + 6

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`

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

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.