Step by step
The question
Question: find f(g(x)) and g(f(x)) for f(x) = x + 4 and g(x) = 2x.
1 Find f(g(x))
f(g(x)) is a function inside a function.
Like with brackets, we start inside.
f works on what g(x) gives.
So first g(x).
For example, with 2: g goes first and gives 4.
Then f works on 4 and gives 8.
g(x) is 2x.
So f(g(x)) is f(2x).
Put (2x) in place of every x in f.
We keep it in brackets.
The + 4 of f just waits at the end.
A bracket with a plus or nothing in front can just be removed.
Now we have 2x + 4.
2 Find g(f(x))
g(f(x)) is a function inside a function.
Like with brackets, we start inside.
g works on what f(x) gives.
So first f(x).
f(x) is x + 4.
So g(f(x)) is g(x + 4).
Put (x + 4) in place of every x in g.
We keep it in brackets: the 2 multiplies the whole x + 4, not just a part.
The 2 in front of the bracket multiplies every part inside.
So we write 2 times each part.
2 times x is 2x.
2 times 4 is 8.
3 Compare
2x + 4 and 2x + 8 are not the same.
So the order matters!
Careful!
Inside first, then outside.
Swap them, and you usually get a different rule.
4 Check
Let's check with a number: x = 1.
Any number works; 1 keeps it easy.
g of 1 is 2, then f of 2 is 6.
f of 1 is 5, then g of 5 is 10.
Different, as we found.
And our rules give the same numbers: 2x + 4 gives 6...
...and 2x + 8 gives 10.
They are right!
Let's look back.
First: find f(g(x)).
Then: find g(f(x)).
Then: compare.
Last: check.