Step by step
The question
Question: find (f/g)(2) for f(x) = x² − 3 and g(x) = 2x + 1.
f(x) = x2 − 3
g(x) = 2x + 1
(f/g)(x) means f of x divided by g of x.
We write it as a fraction: f on top, g at the bottom.
(f/g)(x) = f(x)g(x)
1 Work out f(2)
Put 2 in place of x in f.
f(2) = 22 − 3
Powers come before minus.
Order: ( ) → x² √ → × ÷ → + −
2² means 2 times 2.
22 = 2 × 2 = 4
That is 4.
= 4 − 3
4 minus 3 is 1.
f(2) = 1
2 Work out g(2)
Put 2 in place of x in g.
We write it in brackets.
g(2) = 2(2) + 1
Times comes before plus.
Order: ( ) → x² √ → × ÷ → + −
2(2) = 2 × 2
2(2) means 2 times 2.
2 times 2 is 4.
= 4 + 1
4 plus 1 is 5.
g(2) = 5
3 Divide them
Careful!
Before we divide, look at the bottom.
If g(2) were 0, we could not divide at all.
Careful: the bottom g can never be 0
f(2)/g(2): 1/5.
So (f/g)(2) = 1/5.
(f/g)(2) = 15
4 Check
Let's check another way.
The whole rule: (f/g)(x) = (x² − 3)/(2x + 1).
(f/g)(x) = x2 − 32x + 1
Put in 2: 1/5 again.
It works!
(f/g)(2) = 22 − 32(2) + 1 = 15 = 15
Let's look back.
First: work out f(2).
Then: work out g(2).
Then: divide them.
Last: check.
Answer
(f/g)(2) = 15