Step by step
The question
Question: find the range of f(x) = x² − 5.
f(x) = x2 − 5
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 Take away the 5
In f(x), − 5 comes after.
So we take away 5.
We do it on both sides, so the sign stays true.
x2 − 5 ≥ 0 − 5
0 minus 5 is −5.
The left side is f(x), the y.
So y ≥ −5.
The range is every y from −5 up.
y ≥ −5
3 See it on the graph
On the graph, the turning point is (0, −5).
The curve never goes below y = −5.
Its lowest y is −5.
4 Check
Let's check.
Put in x = 0, the turning point.
We get −5, the smallest y.
x = 0: y = 02 − 5 = 0 − 5 = −5
Now x = 2.
We get −1.
−1 is more than −5.
It fits.
x = 2: y = 22 − 5 = 4 − 5 = −1
Let's look back.
First: what a square can be.
Then: take away the 5.
Then: see it on the graph.
Last: check.
Answer
y ≥ −5