Step by step
The question
Question: find f(g(x)) and g(f(x)) for f(x) = x + 5 and g(x) = 4x.
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 4x.
So f(g(x)) is f(4x).
Put (4x) 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 4x + 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 + 5.
So g(f(x)) is g(x + 5).
Put (x + 5) in place of every x in g.
We keep it in brackets.
The 4 in front of the bracket multiplies every part inside.
So we write 4 times each part.
4 times x is 4x.
4 times 5 is 20.
3 Compare
4x + 5 and 4x + 20 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.
g of 1 is 4, then f of 4 is 9.
f of 1 is 6, then g of 6 is 24.
Different, as we found.
And our rules give the same numbers: 4x + 5 gives 9, 4x + 20 gives 24.
They are right!
Let's look back.
First: find f(g(x)).
Then: find g(f(x)).
Then: compare.
Last: check.