Step by step
The question
Question: find f(2) 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!
2 is an edge: two rules meet there.
It belongs to the rule with the "or equal to" sign.
Careful: x = 2 goes where the "or equal to" is
1 Find the right piece
Let's see where each rule works on a number line.
So we use this rule: 3.
2 Read the rule
This rule has no x.
It always gives 3.
So f(2) = 3.
f(2) = 3
3 Check
Let's check: does 2 fit each rule's x's?
Only one must say yes.
first rule: 2 < −2 no ✗
second rule: 2 ≥ −2 and 2 < 2 no ✗
third rule: 2 ≥ 2 yes
Only the third rule fits.
So f(2) is right!
Let's look back.
First: find the right piece.
Then: read the rule.
Last: check.
Answer
f(2) = 3