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

Step by step

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 times g of x.

We multiply 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!

Times is not putting one inside the other.

We multiply the two rules.

Careful: (f × g)(x) is not f(g(x))

2 Split the first bracket

Each part of the first bracket multiplies the whole second bracket.

(a + b)(c + d) = a(c + d) + b(c + d)

The parts of the first bracket are 2x and 1.

Each one takes the whole second bracket.

= 2x(x − 3) + 1(x − 3)

3 Open the brackets

What is in front of a bracket multiplies every part inside.

a(b + c) = ab + ac

So we write each part times what is in front.

= 2x(x) + 2x(−3) + 1(x) + 1(−3)

− × + = −

Watch the signs.

2x times x is 2x².

2x times (−3) is −6x.

1 times x is x.

1 times (−3) is −3.

= 2x² − 6x + x − 3

4 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.

Each colour is one kind.

like terms = same letters,

same powers

−6x + x is −5x.

−6x + x = −5x

So the expression is 2x² − 5x − 3.

(f × g)(x) = 2x² − 5x − 3

5 Check

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

f of 2 is 5, g of 2 is −1.

Times: −5.

f(2) × g(2) = 5 × (−1) = −5

Now our answer with x = 2.

We get −5 too.

It works!

Answer: 2(2)² − 5(2) − 3 = 8 − 10 − 3 = −5

Let's look back.

First: put the rules in.

Then: split the first bracket.

Then: open the brackets.

Then: group like terms.

Last: check.

Answer