Step by step
The question
Question: find the domain of f(x) = 1/(x² − 4).
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 − 4 zero?
There is an x², so there can be two.
We solve x² − 4 = 0 with the quadratic lesson.
Careful!
x² = 4 has two answers: 2 and −2.
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 − 4, we add 4 on both sides.
0 + 4 = 4.
Now we have x² = 4.
3 Square root of both sides
x squared is 4.
So x is the square root of 4, or minus it.
Minus times minus is plus, so the negative one works too.
2 times 2 is 4.
So x = 2 or x = −2.
4 Write the domain
Both are not allowed.
The domain is all real numbers except −2 and 2: x ≠ −2 and x ≠ 2.
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 2 in: the bottom is 0.
Not allowed.
Now 0: the bottom is −4, 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.