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.
▶ Watch it solved
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.
▶ Watch it solved
The question
Question: graph f(x): x² if x < 1, 4 − x if x ≥ 1.
f(x) = x2 if x < 1
= 4 − x 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!
At an edge, look at the sign.
Less than or greater than: the edge is not in, an open dot.
With "or equal to": a filled dot.
Careful: < >: open dot, ≤ ≥: filled dot
1 Make a table
Each rule gets its own x's, in its own colour.
The edge 1 we work out with both rules: it shows where each piece ends.
2 Draw the grid
Draw the grid.
3 Draw the first piece
x² is for x less than 1.
Plot its points.
At the edge 1: 1 is not in, so an open dot at (1, 1).
Join them with a curve, only for these x's.
It keeps going on the open side.
4 Draw the second piece
4 − x is for x greater than or equal to 1.
Plot its points.
At the edge 1: 1 is in, so a filled dot at (1, 3).
Join them with a straight line, only for these x's.
It keeps going on the open side.
5 Check
Let's check the edge x = 1.
Every x has only one y, so only one dot there can be filled.
1 belongs to the second rule.
f(1) = 3: the filled dot (1, 3).
It works!
f(1) = 4 − 1 = 3
Let's look back.
First: make a table.
Then: draw the grid.
Then: draw the first piece.
Then: draw the second piece.
Last: check.
▶ Watch it solved
