Step by step
The question
Question: find the domain of f(x) = (x + 1)/(x − 5).
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 − 5 = 0.
2 Remove the − 5
We want x by itself.
The − 5 is in the way.
To remove − 5, we add 5.
We do it on both sides, so they stay equal.
− 5 and + 5 make zero.
Cross them out.
0 + 5 = 5.
So x = 5!
3 Write the domain
x = 5 is the only x that is not allowed.
The domain is all real numbers except 5: x ≠ 5.
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 5 in: the bottom is 0.
Dividing by 0 is not allowed.
Now 6: the bottom is 1, not 0.
It works.
Let's look back.
First: find where the bottom is 0.
Then: remove the − 5.
Then: write the domain.
Then: show it on a number line.
Last: check.