Step by step
The question
Question: find (f × g)(x) for f(x) = 2x and g(x) = x + 4.
f(x) = 2x
g(x) = x + 4
(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)(x + 4).
(f × g)(x) = (2x)(x + 4)
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.
Each one takes the whole second bracket.
= 2x(x + 4)
3 Open the brackets
The 2x in front of the bracket multiplies every part inside.
a(b + c) = ab + ac
So we write 2x times each part.
= 2x(x) + 2x(4)
2x times x is 2x².
2x times 4 is 8x.
(f × g)(x) = 2x2 + 8x
4 Check
Let's check with a number: x = 2.
f of 2 is 4, g of 2 is 6.
Times: 24.
f(2) × g(2) = 4 × 6 = 24
Now our answer with x = 2.
We get 24 too.
It works!
Answer: 2(2)2 + 8(2) = 8 + 16 = 24
Let's look back.
First: put the rules in.
Then: split the first bracket.
Then: open the brackets.
Last: check.
Answer
(f × g)(x) = 2x2 + 8x