Step by step
The question
Question: find f(0) for f(x) = 6 − x if x < 3, 2x − 4 if x ≥ 3.
f(x) = 6 − x if x < 3
= 2x − 4 if x ≥ 3
This is a piecewise function.
It has 2 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 greater than or equal to 3, and 0 is not.
Careful: not f(0) = 2(0) − 4 = −4 ✗
1 Find the right piece
Let's see where each rule works on a number line.
So we use this rule: 6 − x.
2 Put 0 in
Put 0 in place of x in 6 − x.
f(0) = 6 − 0
6 minus 0 is 6.
f(0) = 6
So f(0) = 6.
3 Check
Let's check: does 0 fit each rule's x's?
Only one must say yes.
first rule: 0 < 3 yes
second rule: 0 ≥ 3 no ✗
Only the first 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) = 6