Step by step
The question
Question: find (f/g)(x) for f(x) = x − 6 and g(x) = 2x + 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: (x − 6)/(2x + 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 = −2 makes the bottom 0, so we must say it is not allowed.
Which x makes 2x + 4 zero?
We solve 2x + 4 = 0.
We want x by itself.
Two things are stuck to x: times 2 and + 4.
The + 4 is on the outside, so it comes off first.
Then the times 2.
3 Remove the + 4
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.
Now we have 2x = −4.
4 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
−4 divided by 2 is −2.
So x = −2!
So (f/g)(x) = (x − 6)/(2x + 4), with x ≠ −2.
5 Check
Let's check with a number: x = 1.
f of 1 is −5, g of 1 is 6.
Divided: −5/6.
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.
Then: remove the 2.
Last: check.