Step by step
The question
Question: find the domain of f(x) = 5/(2x − 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 of the fraction can never be 0.
Which x makes the bottom 0?
We solve 2x − 6 = 0.
We want x by itself.
Two things are stuck to x: times 2 and − 6.
The − 6 is on the outside, so it comes off first.
Then the times 2.
2 Remove the − 6
To remove − 6, we add 6.
We do it on both sides, so they stay equal.
− 6 and + 6 make zero.
Cross them out.
0 + 6 = 6.
Now we have 2x = 6.
3 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
6 divided by 2 is 3.
So x = 3!
4 Write the domain
x = 3 is the only x that is not allowed.
The domain is all real numbers except 3: x ≠ 3.
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 3 in: the bottom is 0.
Dividing by 0 is not allowed.
Now 4: the bottom is 2, not 0.
It works.
Let's look back.
First: find where the bottom is 0.
Then: remove the − 6.
Then: remove the 2.
Then: write the domain.
Then: show it on a number line.
Last: check.