Step by step
The question
Question: find the domain of f(x) = √(x − 2).
The domain is all the x's we can put in.
1 The inside is 0 or more
No real number times itself is negative.
3 times 3 is 9.
Minus 3 times minus 3 is 9 too.
So we can't take the square root of a negative number.
So the inside, x − 2, must be 0 or more.
2 Remove the − 2
We want x by itself.
The − 2 is in the way.
To remove − 2, we add 2.
We do it on both sides, so the sign stays true.
− 2 and + 2 make zero.
Cross them out.
0 + 2 = 2.
So x is greater than or equal to 2.
These are the x's that can go in.
So the domain is x ≥ 2.
3 Show it on a number line
Books also write the answer like this.
It is called interval notation.
A square bracket means the end is in, like the filled circle.
∞ means it goes on for ever.
4 Check
Let's check.
Put 1 in, just outside.
The inside is −1: negative.
Not allowed.
Now 3, inside: the inside is 1, and the square root of 1 is 1.
It works.
Let's look back.
First: the inside is 0 or more.
Then: remove the − 2.
Then: show it on a number line.
Last: check.