Step by step
The question
Question: find f(0) for f(x) = −1 if x < −2, x + 1 if −2 ≤ x < 2, 3 if x ≥ 2.
f(x) = −1 if x < −2
= x + 1 if −2 ≤ x < 2
= 3 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 first rule!
It is only for x less than −2, and 0 is not.
Careful: not f(0) = −1 ✗
1 Find the right piece
Let's see where each rule works on a number line.
So we use this rule: x + 1.
2 Put 0 in
Put 0 in place of x in x + 1.
f(0) = 0 + 1
0 plus 1 is 1.
f(0) = 1
So f(0) = 1.
3 Check
Let's check: does 0 fit each rule's x's?
Only one must say yes.
first rule: 0 < −2 no ✗
second rule: 0 ≥ −2 and 0 < 2 yes
third rule: 0 ≥ 2 no ✗
Only the second rule fits.
So f(0) is right!
Let's look back.
First: find the right piece.
Then: put 0 in.
Last: check.
Answer
f(0) = 1