Step by step
The question
Question: find f(g(x)) and g(f(x)) for f(x) = 2x and g(x) = 3x.
1 Find f(g(x))
f(g(x)) is a function inside a function.
Like with brackets, we start inside.
First g(x).
g(x) is 3x.
So f(g(x)) is f(3x).
Put (3x) in place of every x in f.
We keep it in brackets.
The 2 in front of the bracket multiplies every part inside.
So we write 2 times each part.
2 times 3x is 6x.
2 Find g(f(x))
g(f(x)) is a function inside a function.
Like with brackets, we start inside.
First f(x).
f(x) is 2x.
So g(f(x)) is g(2x).
Put (2x) in place of every x in g.
We keep it in brackets.
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
3 times 2x is 6x.
3 Compare
Both are 6x.
Here the order did not matter.
But usually it does.
Careful!
Inside first, then outside.
Swap them, and you usually get a different rule.
4 Check
Let's check with a number: x = 1.
g of 1 is 3, then f of 3 is 6.
f of 1 is 2, then g of 2 is 6.
The same, as we found.
And our rules give the same numbers: 6x gives 6, 6x gives 6.
They are right!
Let's look back.
First: find f(g(x)).
Then: find g(f(x)).
Then: compare.
Last: check.