Step by step
The question
Question: find f(−3) for f(x) = x² − 4 if x < −1, 3x if −1 ≤ x ≤ 2, 7 − x if x > 2.
f(x) = x2 − 4 if x < −1
= 3x if −1 ≤ x ≤ 2
= 7 − x if x > 2
This is a piecewise function.
It has 3 rules.
Which rule we use depends on x.
Each x uses only one rule.
piecewise: a different rule for different x's
each x uses only one rule
Careful!
Not the second rule!
It is only for x from −1 to 2, and −3 is not.
Careful: not f(−3) = 3(−3) = −9 ✗
1 Find the right piece
Let's see where each rule works on a number line.
So we use this rule: x² − 4.
2 Put −3 in
Put −3 in place of x in x² − 4.
We write it in brackets.
f(−3) = (−3)2 − 4
Powers come before minus.
Order: ( ) → x² √ → × ÷ → + −
(−3)² means −3 times −3.
Minus times minus is plus.
(−3)2 = (−3) × (−3) = 9
That is 9.
= 9 − 4
9 minus 4 is 5.
f(−3) = 5
So f(−3) = 5.
3 Check
Let's check: does −3 fit each rule's x's?
Only one must say yes.
first rule: −3 < −1 yes
second rule: −3 ≥ −1 and −3 ≤ 2 no ✗
third rule: −3 > 2 no ✗
Only the first rule fits.
So f(−3) is right!
Let's look back.
First: find the right piece.
Then: put −3 in.
Last: check.
Answer
f(−3) = 5