Step by step
The question
Question: find f(g(x)) and g(f(x)) for f(x) = x + 1 and g(x) = x + 4.
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 x + 4.
So f(g(x)) is f(x + 4).
Put (x + 4) in place of every x in f.
We keep it in brackets.
A bracket with a plus or nothing in front can just be removed.
Now we have x + 4 + 1.
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
Plain numbers go together.
Each colour is one kind.
4 + 1 is 5.
So the expression is x + 5.
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 x + 1.
So g(f(x)) is g(x + 1).
Put (x + 1) in place of every x in g.
We keep it in brackets.
A bracket with a plus or nothing in front can just be removed.
Now we have x + 1 + 4.
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
Plain numbers go together.
Each colour is one kind.
1 + 4 is 5.
So the expression is x + 5.
3 Compare
Both are x + 5.
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 5, then f of 5 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: x + 5 gives 6, x + 5 gives 6.
They are right!
Let's look back.
First: find f(g(x)).
Then: find g(f(x)).
Then: compare.
Last: check.