Step by step
The question
Question: find the domain of f(x) = √(3x − 12).
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, 3x − 12, must be 0 or more.
We want x by itself.
Two things are stuck to x: times 3 and − 12.
The − 12 is on the outside, so it comes off first.
Then the times 3.
2 Remove the − 12
To remove − 12, we add 12.
We do it on both sides, so the sign stays true.
− 12 and + 12 make zero.
Cross them out.
0 + 12 = 12.
Now we have 3x ≥ 12.
3 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so the sign stays true.
3 divided by 3 is 1.
So just x is left.
12 divided by 3 is 4.
So x is greater than or equal to 4.
These are the x's that can go in.
So the domain is x ≥ 4.
4 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.
5 Check
Let's check.
Put 3 in, just outside.
The inside is −3: negative.
Not allowed.
Now 7, inside: the inside is 9, and the square root of 9 is 3.
It works.
Let's look back.
First: the inside is 0 or more.
Then: remove the − 12.
Then: remove the 3.
Then: show it on a number line.
Last: check.