Step by step
The question
Question: find the domain of f(x) = 1/(2x − 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 2x − 3 = 0.
We want x by itself.
Two things are stuck to x: times 2 and − 3.
The − 3 is on the outside, so it comes off first.
Then the times 2.
2 Remove the − 3
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.
Now we have 2x = 3.
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.
So x = 3/2!
4 Write the domain
x = 3/2 is the only x that is not allowed.
The domain is all real numbers except 3/2: x ≠ 3/2.
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.
Let's look back.
First: find where the bottom is 0.
Then: remove the − 3.
Then: remove the 2.
Then: write the domain.
Last: show it on a number line.