Step by step
The question
Question: find (f/g)(x) for f(x) = 2x − 1 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: (2x − 1)/(x + 4).
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 = −4 makes the bottom 0, so we must say it is not allowed.
Which x makes x + 4 zero?
We solve x + 4 = 0.
3 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) = (2x − 1)/(x + 4), with x ≠ −4.
4 Check
Let's check with a number: x = 1.
f of 1 is 1, g of 1 is 5.
Divided: 1/5.
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 + 4.
Last: check.