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