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

Step by step

The question

Question: find (f − g)(x) for f(x) = x² + 3x and g(x) = x² − 1.

f(x) = x² + 3x

g(x) = x² − 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: (x² + 3x) − (x² − 1).

(f − g)(x) = (x² + 3x) − (x² − 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 x² + 3x − x² − 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 x² + 3x − x² + 1.

= x² + 3x − x² + 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 x² terms 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.

= x² − x² + 3x + 1

x² − x² is 0.

It is like 1 apple take away 1 apple: that is 0 apples.

x² − x² = 0

So the expression is 3x + 1.

(f − g)(x) = 3x + 1

4 Check

Let's check with a number: x = 2.

f of 2 is 10, g of 2 is 3.

Minus: 7.

f(2) − g(2) = 10 − 3 = 7

Now our answer with x = 2.

We get 7 too.

It works!

Answer: 3(2) + 1 = 6 + 1 = 7

Let's look back.

First: put the rules in.

Then: open the brackets.

Then: group like terms.

Last: check.

Answer