Worked examples
The question
Question: find (f + g)(x) for f(x) = 2x + 1 and g(x) = x − 3.
f(x) = 2x + 1
g(x) = x − 3
(f + g)(x) means f of x plus g of x.
We add the two answers.
(f + g)(x) = f(x) + g(x)
1 Put the rules in
Each rule goes in brackets, as one piece: (2x + 1) + (x − 3).
(f + g)(x) = (2x + 1) + (x − 3)
2 Open the brackets
A bracket with a plus or nothing in front can just be removed.
( ) away
Now we have 2x + 1 + x − 3.
= 2x + 1 + x − 3
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.
like terms = same letters,
same powers
We move like terms next to each other.
Each term takes its sign with it.
= 2x + x + 1 − 3
2x + x is 3x.
It is like 2 apples plus 1 apple: that is 3 apples.
1 − 3 is −2.
2x + x = 3x
1 − 3 = −2
So the expression is 3x − 2.
(f + g)(x) = 3x − 2
4 Check
Let's check with a number: x = 2.
f of 2 is 5, g of 2 is −1.
Plus: 4.
f(2) + g(2) = 5 + (−1) = 4
Now our answer with x = 2.
We get 4 too.
It works!
Answer: 3(2) − 2 = 6 − 2 = 4
Let's look back.
First: put the rules in.
Then: open the brackets.
Then: group like terms.
Last: check.
Answer (f + g)(x) = 3x − 2
▶ Watch it solved
The question
Question: find (f − g)(x) for f(x) = 2x + 1 and g(x) = x − 3.
f(x) = 2x + 1
g(x) = x − 3
(f − g)(x) means f of x minus g of x.
We take away the two answers.
(f − g)(x) = f(x) − g(x)
1 Put the rules in
Each rule goes in brackets, as one piece: (2x + 1) − (x − 3).
(f − g)(x) = (2x + 1) − (x − 3)
Careful!
The minus is in front of the whole of g(x).
It changes every sign in the bracket, not only the first.
Careful: not 2x + 1 − x − 3 ✗
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 2x + 1 − x + 3.
= 2x + 1 − x + 3
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.
like terms = same letters,
same powers
We move like terms next to each other.
Each term takes its sign with it.
= 2x − x + 1 + 3
2x − x is x.
It is like 2 apples take away 1 apple: that is 1 apple.
1 + 3 is 4.
2x − x = x
1 + 3 = 4
So the expression is x + 4.
(f − g)(x) = x + 4
4 Check
Let's check with a number: x = 2.
f of 2 is 5, g of 2 is −1.
Minus: 6.
f(2) − g(2) = 5 − (−1) = 6
Now our answer with x = 2.
We get 6 too.
It works!
Answer: 2 + 4 = 6
Let's look back.
First: put the rules in.
Then: open the brackets.
Then: group like terms.
Last: check.
Answer (f − g)(x) = x + 4
▶ Watch it solved
The question
Question: find (f × g)(−1) for f(x) = x + 4 and g(x) = 2x − 1.
f(x) = x + 4
g(x) = 2x − 1
(f × g)(x) means f of x times g of x.
We multiply the two answers.
(f × g)(x) = f(x) × g(x)
1 Work out f(−1)
Put −1 in place of x in f.
f(−1) = −1 + 4
Start at −1.
Go up 4.
We land on 3.
f(−1) = 3
2 Work out g(−1)
Put −1 in place of x in g.
We write it in brackets.
g(−1) = 2(−1) − 1
Times comes before minus.
Order: ( ) → x² √ → × ÷ → + −
2(−1) = 2 × (−1)
+ × − = −
2(−1) means 2 times −1.
Plus times minus is minus.
2 times 1 is 2.
So we get −2.
= −2 − 1
Start at −2.
Go down 1.
We land on −3.
g(−1) = −3
3 Multiply them
f(−1) times g(−1): 3 × (−3).
(f × g)(−1) = 3 × (−3)
So (f × g)(−1) = −9.
(f × g)(−1) = −9
4 Check
Let's check another way.
First the whole rule: (f × g)(x) = 2x² + 7x − 4.
(f × g)(x) = 2x2 + 7x − 4
Then put in −1.
We get −9 again.
It works!
(f × g)(-1) = 2(−1)2 + 7(−1) − 4 = 2 − 7 − 4
= −9
Let's look back.
First: work out f(−1).
Then: work out g(−1).
Then: multiply them.
Last: check.
▶ Watch it solved
