Step by step
The question
Question: find f(g(x)) and g(f(x)) for f(x) = 3x + 2 and g(x) = x − 1.
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 − 1.
So f(g(x)) is f(x − 1).
Put (x − 1) in place of every x in f.
We keep it in brackets.
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
Watch the signs.
3 times x is 3x.
3 times (−1) is −3.
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.
−3 + 2 is −1.
So the expression is 3x − 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 3x + 2.
So g(f(x)) is g(3x + 2).
Put (3x + 2) 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 3x + 2 − 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.
2 − 1 is 1.
So the expression is 3x + 1.
3 Compare
3x − 1 and 3x + 1 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 0, then f of 0 is 2.
f of 1 is 5, then g of 5 is 4.
Different, as we found.
And our rules give the same numbers: 3x − 1 gives 2, 3x + 1 gives 4.
They are right!
Let's look back.
First: find f(g(x)).
Then: find g(f(x)).
Then: compare.
Last: check.