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