f(x) = 3 - x, g(x) = 2x + 1, find (f

Step by step

The question

Question: find (f − g)(x) for f(x) = 3 − x and g(x) = 2x + 1.

f(x) = 3 − x

g(x) = 2x + 1

(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: (3 − x) − (2x + 1).

(f − g)(x) = (3 − x) − (2x + 1)

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 3 − x − 2x + 1 ✗

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 3 − x − 2x − 1.

= 3 − x − 2x − 1

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.

= −x − 2x + 3 − 1

3 − 1 is 2.

−x − 2x is −3x.

3 − 1 = 2

−x − 2x = −3x

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 1, g of 2 is 5.

Minus: −4.

f(2) − g(2) = 1 − 5 = −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