Step by step
The question
Question: find the domain and range of f(x) = √(x + 4).
The domain is all the x's we can put in.
The range is all the answers, the y's, that can come out.
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 + 4, must be 0 or more.
2 Remove the + 4
We want x by itself.
The + 4 is in the way.
To remove + 4, we take away 4.
We do it on both sides, so the sign stays true.
+ 4 and − 4 make zero.
Cross them out.
0 − 4 = −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.
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.
Now the range: which y's can come out?
4 What a square root can be
A square root is never below 0.
The square root of 0 is 0.
The square root of 4 is 2.
So √(x + 4) is 0 or more.
The left side is the y.
So y ≥ 0.
The range is every y from 0 up.
5 Check
Let's check.
Put in x = −4: the inside is 0.
We get 0, the smallest y.
Let's look back.
First: the inside is 0 or more.
Then: remove the + 4.
Then: show it on a number line.
Then: what a square root can be.
Last: check.