Step by step
The question
Question: find the domain of f(x) = 1/(x² + x − 6).
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 + x − 6 zero?
There is an x², so there can be two.
We solve x² + x − 6 = 0 with the quadratic lesson.
Careful!
The bottom is 0 when either bracket is 0.
So two x's are not allowed.
2 Factor
We need two numbers.
They multiply to −6, and they add to 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.
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.
So it is (x − 2)(x + 3) = 0.
3 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
We make each bracket 0.
Two small equations.
In x − 2 = 0, we add 2 on both sides.
In x + 3 = 0, we take away 3 on both sides.
So x = 2 or x = −3.
4 Write the domain
Both are not allowed.
The domain is all real numbers except −3 and 2: x ≠ −3 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 −6, not 0.
It works.
Let's look back.
First: find where the bottom is 0.
Then: factor.
Then: one bracket is 0.
Then: write the domain.
Then: show it on a number line.
Last: check.