Step by step
The question
Question: find (f/g)(x) for f(x) = x² + x − 6 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 + x − 6)/(x − 2).
2 Simplify
The top: x² + x − 6.
We need two numbers.
They multiply to −6, and they add to 1.
They multiply to a minus number, so one number is plus and one is minus.
They add to a plus number, so the bigger one is plus.
We list the pairs that make 6, smallest first.
We take the smaller from the bigger.
We need 1.
2 and 3 give 1.
Yes!
With the signs: −2 and 3.
So the top is (x − 2)(x + 3).
Now the same bracket is on the top and the bottom.
(x − 2)/(x − 2) is 1.
Cross them out.
What is left: x + 3.
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 add 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 + 3, with x ≠ 2.
5 Check
Let's check with a number: x = 1.
f of 1 is −4, g of 1 is −1.
Divided: 4.
Now our answer with x = 1: 4 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.