Step by step
The question
Question: find the domain of f(x) = 3/(x² + 4x).
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 + 4x zero?
There is an x², so there can be two.
We solve x² + 4x = 0 with the quadratic lesson.
Careful!
The bottom is 0 when either bracket is 0.
So two x's are not allowed.
2 Factor
There is no plain number.
Both parts have x in them.
So we take x out.
So it is x(x + 4) = 0.
3 One bracket is 0
x times the bracket is 0.
That only happens if one of them is 0.
We make each part 0.
Two small equations.
x = 0 is done already.
In x + 4 = 0, we take away 4 on both sides.
So x = 0 or x = −4.
4 Write the domain
Both are not allowed.
The domain is all real numbers except −4 and 0: x ≠ −4 and x ≠ 0.
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 0 in: the bottom is 0.
Not allowed.
Now 1: the bottom is 5, 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.