Step by step
The question
Question: find f(4) for f(x) = x + 2 if x < 1, 3x − 1 if x ≥ 1.
f(x) = x + 2 if x < 1
= 3x − 1 if x ≥ 1
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 first rule!
It is only for x less than 1, and 4 is not.
Careful: not f(4) = 4 + 2 = 6 ✗
1 Find the right piece
Let's see where each rule works on a number line.
So we use this rule: 3x − 1.
2 Put 4 in
Put 4 in place of x in 3x − 1.
We write it in brackets.
f(4) = 3(4) − 1
Times comes before minus.
Order: ( ) → x² √ → × ÷ → + −
3(4) = 3 × 4
3(4) means 3 times 4.
3 times 4 is 12.
= 12 − 1
12 minus 1 is 11.
f(4) = 11
So f(4) = 11.
3 Check
Let's check: does 4 fit each rule's x's?
Only one must say yes.
first rule: 4 < 1 no ✗
second rule: 4 ≥ 1 yes
Only the second rule fits.
So f(4) is right!
Let's look back.
First: find the right piece.
Then: put 4 in.
Last: check.
Answer
f(4) = 11