Step by step
The question
Question: find f(1) 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!
1 is an edge: two rules meet there.
It belongs to the rule with the "or equal to" sign.
Careful: x = 1 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: 3x − 1.
2 Put 1 in
Put 1 in place of x in 3x − 1.
We write it in brackets.
f(1) = 3(1) − 1
Times comes before minus.
Order: ( ) → x² √ → × ÷ → + −
3(1) = 3 × 1
3(1) means 3 times 1.
3 times 1 is 3.
= 3 − 1
3 minus 1 is 2.
f(1) = 2
So f(1) = 2.
3 Check
Let's check: does 1 fit each rule's x's?
Only one must say yes.
first rule: 1 < 1 no ✗
second rule: 1 ≥ 1 yes
Only the second rule fits.
So f(1) is right!
Let's look back.
First: find the right piece.
Then: put 1 in.
Last: check.
Answer
f(1) = 2