Step by step
The question
Question: find (f/g)(x) for f(x) = x² − 2x − 8 and g(x) = x + 2.
(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 − 2x − 8)/(x + 2).
2 Simplify
The top: x² − 2x − 8.
We need two numbers.
They multiply to −8, and they add to −2.
They multiply to a minus number, so one number is plus and one is minus.
They add to a minus number, so the bigger one is minus.
We list the pairs that make 8, smallest first.
We take the smaller from the bigger.
We need 2.
2 and 4 give 2.
Yes!
With the signs: 2 and −4.
So the top is (x + 2)(x − 4).
Now the same bracket is on the top and the bottom.
(x + 2)/(x + 2) 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 = −2 still makes the bottom 0.
So it stays not allowed.
Which x makes x + 2 zero?
We solve x + 2 = 0.
4 Remove the + 2
We want x by itself.
The + 2 is in the way.
To remove + 2, we take away 2.
We do it on both sides, so they stay equal.
+ 2 and − 2 make zero.
Cross them out.
0 − 2 = −2.
So x = −2!
So (f/g)(x) = x − 4, with x ≠ −2.
5 Check
Let's check with a number: x = 1.
f of 1 is −9, g of 1 is 3.
Divided: −3.
Now our answer with x = 1: −3 too.
It works!
Let's look back.
First: put the rules in.
Then: simplify.
Then: find what x can't be.
Then: remove the + 2.
Last: check.