Step by step
The question
Question: find f(g(x)) and g(f(x)) for f(x) = 4x + 1 and g(x) = 2x.
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 2x.
So f(g(x)) is f(2x).
Put (2x) in place of every x in f.
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 2x is 8x.
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 4x + 1.
So g(f(x)) is g(4x + 1).
Put (4x + 1) in place of every x in g.
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 4x is 8x.
2 times 1 is 2.
3 Compare
8x + 1 and 8x + 2 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 2, then f of 2 is 9.
f of 1 is 5, then g of 5 is 10.
Different, as we found.
And our rules give the same numbers: 8x + 1 gives 9, 8x + 2 gives 10.
They are right!
Let's look back.
First: find f(g(x)).
Then: find g(f(x)).
Then: compare.
Last: check.