Step by step
The question
Question: find the domain of f(x) = √(6 − 2x).
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, 6 − 2x, must be 0 or more.
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 take away 6.
We do it on both sides, so the sign stays true.
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.
We divide by a negative number.
That turns the sign around.
Greater than or equal to becomes less than or equal to.
−2 divided by −2 is 1.
So just x is left.
−6 divided by −2 is 3.
So x is less than or equal to 3.
Careful!
Many people forget to turn the sign.
They get x ≥ 3.
But we divided by a negative number.
These are the x's that can go in.
So the domain is x ≤ 3.
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 4 in, just outside.
The inside is −2: negative.
Not allowed.
Now 1, inside: the inside is 4, and the square root of 4 is 2.
It works.
Let's look back.
First: the inside is 0 or more.
Then: remove the 6.
Then: remove the −2.
Then: show it on a number line.
Last: check.