Step by step
The question
Question: find f(g(x)) and g(f(x)) for f(x) = x − 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.
A bracket with a plus or nothing in front can just be removed.
Now we have 2x − 1.
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.
The 2 in front of the bracket multiplies every part inside.
So we write 2 times each part.
Watch the signs.
2 times x is 2x.
2 times (−1) is −2.
3 Compare
2x − 1 and 2x − 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 1.
f of 1 is 0, then g of 0 is 0.
Different, as we found.
And our rules give the same numbers: 2x − 1 gives 1, 2x − 2 gives 0.
They are right!
Let's look back.
First: find f(g(x)).
Then: find g(f(x)).
Then: compare.
Last: check.