Worked examples
The question
Question: find (f/g)(x) for f(x) = x + 3 and g(x) = x − 2.
f(x) = x + 3
g(x) = x − 2
(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.
(f/g)(x) = f(x)g(x)
1 Put the rules in
Each rule goes in as one piece: (x + 3)/(x − 2).
(f/g)(x) = x + 3x − 2
2 Find what x can't be
We can never divide by 0.
So the bottom, g of x, can never be 0.
÷ 0: not allowed → g(x) ≠ 0
Careful!
The answer is not only the fraction.
x = 2 makes the bottom 0, so we must say it is not allowed.
Careful: write the x that g can't be
Which x makes x − 2 zero?
We solve x − 2 = 0.
x − 2 = 0
3 Remove the − 2
We want x by itself.
The − 2 is in the way.
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
x − 2 + 2 = 0 + 2
− 2 and + 2 make zero.
Cross them out.
0 + 2 = 2.
0 + 2 = 2
So x = 2!
x = 2
So (f/g)(x) = (x + 3)/(x − 2), with x ≠ 2.
(f/g)(x) = x + 3x − 2, x ≠ 2
4 Check
Let's check with a number: x = 1.
f of 1 is 4, g of 1 is −1.
Divided: −4.
f(1)g(1) = 4−1 = −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 − 2.
Last: check.
Answer (f/g)(x) = x + 3x − 2, x ≠ 2
▶ Watch it solved
The question
Question: find (f/g)(x) for f(x) = x² − 9 and g(x) = x + 3.
f(x) = x2 − 9
g(x) = x + 3
(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.
(f/g)(x) = f(x)g(x)
1 Put the rules in
Each rule goes in as one piece: (x² − 9)/(x + 3).
(f/g)(x) = x2 − 9x + 3
2 Simplify
The top: x² − 9.
top : x2 − 9
x² − 9 is x² minus 3 squared.
A square minus a square is the plus bracket times the minus bracket.
a² − b² = (a + b)(a − b)
So the top is (x + 3)(x − 3).
= (x + 3)(x − 3)
Now the same bracket is on the top and the bottom.
(f/g)(x) = (x + 3)(x − 3)(x + 3)
(x + 3)/(x + 3) is 1.
Cross them out.
What is left: x − 3.
(f/g)(x) = x − 3
3 Find what x can't be
We can never divide by 0.
So the bottom, g of x, can never be 0.
÷ 0: not allowed → g(x) ≠ 0
Careful!
The crossed-out bracket was the bottom, g.
In the question, x = −3 still makes the bottom 0.
So it stays not allowed.
Careful: x ≠ −3 stays, even after crossing out
Which x makes x + 3 zero?
We solve x + 3 = 0.
x + 3 = 0
4 Remove the + 3
We want x by itself.
The + 3 is in the way.
To remove + 3, we take away 3.
We do it on both sides, so they stay equal.
x + 3 − 3 = 0 − 3
+ 3 and − 3 make zero.
Cross them out.
0 − 3 = −3.
0 − 3 = −3
So x = −3!
x = −3
So (f/g)(x) = x − 3, with x ≠ −3.
(f/g)(x) = x − 3, x ≠ −3
5 Check
Let's check with a number: x = 1.
f of 1 is −8, g of 1 is 4.
Divided: −2.
f(1)g(1) = −84 = −2
Now our answer with x = 1: −2 too.
It works!
Answer: 1 − 3 = −2
Let's look back.
First: put the rules in.
Then: simplify.
Then: find what x can't be.
Then: remove the + 3.
Last: check.
Answer (f/g)(x) = x − 3, x ≠ −3
▶ Watch it solved
The question
Question: find (f/g)(x) for f(x) = x² + x − 6 and g(x) = x − 2.
f(x) = x2 + x − 6
g(x) = x − 2
(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.
(f/g)(x) = f(x)g(x)
1 Put the rules in
Each rule goes in as one piece: (x² + x − 6)/(x − 2).
(f/g)(x) = x2 + x − 6x − 2
2 Simplify
The top: x² + x − 6.
top : x2 + x − 6
We need two numbers.
They multiply to −6, and they add to 1.
? × ? = −6, ? + ? = 1
They multiply to a minus number, so one number is plus and one is minus.
They add to a plus number, so the bigger one is plus.
× is − → one +, one −; bigger one +
We list the pairs that make 6, smallest first.
We take the smaller from the bigger.
We need 1.
2 and 3 give 1.
Yes!
With the signs: −2 and 3.
1 × 6: 6 − 1 = 5
2 × 3: 3 − 2 = 1 ✓
So the top is (x − 2)(x + 3).
= (x − 2)(x + 3)
Now the same bracket is on the top and the bottom.
(f/g)(x) = (x − 2)(x + 3)(x − 2)
(x − 2)/(x − 2) is 1.
Cross them out.
What is left: x + 3.
(f/g)(x) = x + 3
3 Find what x can't be
We can never divide by 0.
So the bottom, g of x, can never be 0.
÷ 0: not allowed → g(x) ≠ 0
Careful!
The crossed-out bracket was the bottom, g.
In the question, x = 2 still makes the bottom 0.
So it stays not allowed.
Careful: x ≠ 2 stays, even after crossing out
Which x makes x − 2 zero?
We solve x − 2 = 0.
x − 2 = 0
4 Remove the − 2
We want x by itself.
The − 2 is in the way.
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
x − 2 + 2 = 0 + 2
− 2 and + 2 make zero.
Cross them out.
0 + 2 = 2.
0 + 2 = 2
So x = 2!
x = 2
So (f/g)(x) = x + 3, with x ≠ 2.
(f/g)(x) = x + 3, x ≠ 2
5 Check
Let's check with a number: x = 1.
f of 1 is −4, g of 1 is −1.
Divided: 4.
f(1)g(1) = −4−1 = 4
Now our answer with x = 1: 4 too.
It works!
Answer: 1 + 3 = 4
Let's look back.
First: put the rules in.
Then: simplify.
Then: find what x can't be.
Then: remove the − 2.
Last: check.
Answer (f/g)(x) = x + 3, x ≠ 2
▶ Watch it solved
