Step by step
The question
Question: find (f/g)(x) for f(x) = 3x and g(x) = x − 5.
(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: (3x)/(x − 5).
2 Find what x can't be
We can never divide by 0.
So the bottom, g of x, can never be 0.
Careful!
The answer is not only the fraction.
x = 5 makes the bottom 0, so we must say it is not allowed.
Which x makes x − 5 zero?
We solve x − 5 = 0.
3 Remove the − 5
We want x by itself.
The − 5 is in the way.
To remove − 5, we add 5.
We do it on both sides, so they stay equal.
− 5 and + 5 make zero.
Cross them out.
0 + 5 = 5.
So x = 5!
So (f/g)(x) = (3x)/(x − 5), with x ≠ 5.
4 Check
Let's check with a number: x = 1.
f of 1 is 3, g of 1 is −4.
Divided: −3/4.
That is our fraction at x = 1.
It works!
Let's look back.
First: put the rules in.
Then: find what x can't be.
Then: remove the − 5.
Last: check.