Step by step
The question
Question: find the domain of f(x) = 3/(x² − 9).
The domain is all the x's we can put in.
1 Find where the bottom is 0
We can never divide by 0.
So the bottom can never be 0.
Which x's make x² − 9 zero?
There is an x², so there can be two.
We solve x² − 9 = 0 with the quadratic lesson.
Careful!
x² = 9 has two answers: 3 and −3.
Both make the bottom 0, so both are not allowed.
2 Get x² alone
There is no x term, so we get x² by itself.
To remove the − 9, we add 9 on both sides.
0 + 9 = 9.
Now we have x² = 9.
3 Square root of both sides
x squared is 9.
So x is the square root of 9, or minus it.
Minus times minus is plus, so the negative one works too.
3 times 3 is 9.
So x = 3 or x = −3.
4 Write the domain
Both are not allowed.
The domain is all real numbers except −3 and 3: x ≠ −3 and x ≠ 3.
5 Show it on a number line
Books also write the answer like this.
It is called interval notation.
A round bracket means the end is not in, like the open circle.
∞ means it goes on for ever.
∪ means or.
6 Check
Let's check.
Put 3 in: the bottom is 0.
Not allowed.
Now 0: the bottom is −9, not 0.
It works.
Let's look back.
First: find where the bottom is 0.
Then: get x² alone.
Then: square root of both sides.
Then: write the domain.
Then: show it on a number line.
Last: check.