Step by step
The question
Question: find the domain and range of f(x) = x² − 2.
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 Look for a problem
Two things can go wrong: a 0 at the bottom of a fraction, or a negative number under a square root.
Here there is no fraction with x at the bottom, and no square root.
So any number can go in: the domain is all real numbers.
Now the range: which y's can come out?
2 What a square can be
A square is never below 0.
3 squared is 9.
Minus 3 squared is 9.
0 squared is 0.
So x² is 0 or more.
3 Take away the 2
In f(x), − 2 comes after.
So we take away 2.
We do it on both sides, so the sign stays true.
0 minus 2 is −2.
The left side is f(x), the y.
So y ≥ −2.
The range is every y from −2 up.
4 See it on the graph
On the graph, the turning point is (0, −2).
The curve never goes below y = −2.
Its lowest y is −2.
5 Check
Let's check.
Put in x = 0, the turning point.
We get −2, the smallest y.
Now x = 2.
We get 2.
2 is more than −2.
It fits.
Let's look back.
First: look for a problem.
Then: what a square can be.
Then: take away the 2.
Then: see it on the graph.
Last: check.