Step by step
The question
Question: find (f/g)(x) for f(x) = x² + 7x + 12 and g(x) = x + 3.
(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.
1 Put the rules in
Each rule goes in as one piece: (x² + 7x + 12)/(x + 3).
2 Simplify
The top: x² + 7x + 12.
We need two numbers.
They multiply to 12, and they add to 7.
They multiply to a plus number, so the two numbers have the same sign.
They add to a plus number, so both are plus.
We list the pairs that make 12, smallest first.
We add each pair.
We need 7.
3 and 4 give 7.
Yes!
So the top is (x + 3)(x + 4).
Now the same bracket is on the top and the bottom.
(x + 3)/(x + 3) is 1.
Cross them out.
What is left: x + 4.
3 Find what x can't be
We can never divide by 0.
So the bottom, g of x, can never be 0.
Careful!
The crossed-out bracket was the bottom, g.
In the question, x = −3 still makes the bottom 0.
So it stays not allowed.
Which x makes x + 3 zero?
We solve x + 3 = 0.
4 Remove the + 3
We want x by itself.
The + 3 is in the way.
To remove + 3, we take away 3.
We do it on both sides, so they stay equal.
+ 3 and − 3 make zero.
Cross them out.
0 − 3 = −3.
So x = −3!
So (f/g)(x) = x + 4, with x ≠ −3.
5 Check
Let's check with a number: x = 1.
f of 1 is 20, g of 1 is 4.
Divided: 5.
Now our answer with x = 1: 5 too.
It works!
Let's look back.
First: put the rules in.
Then: simplify.
Then: find what x can't be.
Then: remove the + 3.
Last: check.