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

Step by step

The question

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

f(x) = x + 2

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: (x + 2)(x − 3).

(f × g)(x) = (x + 2)(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 x and 2.

Each one takes the whole second bracket.

= x(x − 3) + 2(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.

= x(x) + x(−3) + 2(x) + 2(−3)

− × + = −

Watch the signs.

x times x is x².

x times (−3) is −3x.

2 times x is 2x.

2 times (−3) is −6.

= x² − 3x + 2x − 6

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

−3x + 2x is −x.

−3x + 2x = −x

So the expression is x² − x − 6.

(f × g)(x) = x² − x − 6

5 Check

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

f of 2 is 4, g of 2 is −1.

Times: −4.

f(2) × g(2) = 4 × (−1) = −4

Now our answer with x = 2.

We get −4 too.

It works!

Answer: 2² − 2 − 6 = 4 − 2 − 6 = −4

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