Step by step
The question
Question: find the range of f(x) = x² + 3.
f(x) = x2 + 3
The range is all the answers, the y's, that can come out.
range = the y's that can come out
1 What a square can be
A square is never below 0.
3 squared is 9.
Minus 3 squared is 9.
0 squared is 0.
3² = 9, (−3)² = 9, 0² = 0
So x² is 0 or more.
x2 ≥ 0
2 Add the 3
In f(x), + 3 comes after.
So we add 3.
We do it on both sides, so the sign stays true.
x2 + 3 ≥ 0 + 3
0 plus 3 is 3.
The left side is f(x), the y.
So y ≥ 3.
The range is every y from 3 up.
y ≥ 3
3 See it on the graph
On the graph, the turning point is (0, 3).
The curve never goes below y = 3.
Its lowest y is 3.
4 Check
Let's check.
Put in x = 0, the turning point.
We get 3, the smallest y.
x = 0: y = 02 + 3 = 0 + 3 = 3
Now x = 2.
We get 7.
7 is more than 3.
It fits.
x = 2: y = 22 + 3 = 4 + 3 = 7
Let's look back.
First: what a square can be.
Then: add the 3.
Then: see it on the graph.
Last: check.
Answer
y ≥ 3