Step by step
The question
Question: find (f/g)(x) for f(x) = x² − 16 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 − 16)/(x + 4).
2 Simplify
The top: x² − 16.
x² − 16 is x² minus 4 squared.
A square minus a square is the plus bracket times the minus bracket.
So the top is (x + 4)(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 − 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 = −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 take away 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 − 4, with x ≠ −4.
5 Check
Let's check with a number: x = 1.
f of 1 is −15, g of 1 is 5.
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 + 4.
Last: check.