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