Step by step
The question
Question: find (f − g)(x) for f(x) = 4x − 1 and g(x) = x + 5.
(f − g)(x) means f of x minus g of x.
We take away the two answers.
1 Put the rules in
Each rule goes in brackets, as one piece: (4x − 1) − (x + 5).
Careful!
The minus is in front of the whole of g(x).
It changes every sign in the bracket, not only the first.
2 Open the brackets
A bracket with a plus or nothing in front can just be removed.
A minus in front of a bracket changes the sign of every part inside.
Now we have 4x − 1 − x − 5.
3 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
Plain numbers go together.
Each colour is one kind.
We move like terms next to each other.
Each term takes its sign with it.
4x − x is 3x.
It is like 4 apples take away 1 apple: that is 3 apples.
−1 − 5 is −6.
So the expression is 3x − 6.
4 Check
Let's check with a number: x = 2.
f of 2 is 7, g of 2 is 7.
Minus: 0.
Now our answer with x = 2.
We get 0 too.
It works!
Let's look back.
First: put the rules in.
Then: open the brackets.
Then: group like terms.
Last: check.