Step by step
The question
Question: find the domain of f(x) = 1/(x − 3).
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 of the fraction can never be 0.
Which x makes the bottom 0?
We solve x − 3 = 0.
2 Remove the − 3
We want x by itself.
The − 3 is in the way.
To remove − 3, we add 3.
We do it on both sides, so they stay equal.
− 3 and + 3 make zero.
Cross them out.
0 + 3 = 3.
So x = 3!
3 Write the domain
x = 3 is the only x that is not allowed.
The domain is all real numbers except 3: x ≠ 3.
4 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.
5 Check
Let's check.
Put 3 in: the bottom is 0.
Dividing by 0 is not allowed.
Now 4: the bottom is 1, not 0.
It works.
Let's look back.
First: find where the bottom is 0.
Then: remove the − 3.
Then: write the domain.
Then: show it on a number line.
Last: check.