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Name:____________________________ Date:_______________ Band:________

Algebra II | Packer Collegiate Institute

Piecewise Functions

Piecewise functions are functions made up of different “pieces.” See, for example, the following graph:

Do you see the two different parts? First part equation: ____________________ Second part equation: __________________ What x-values does the first part cover? (express in interval notation and in inequality form) Interval: Inequality: What x-values does the second part cover? (express in interval notation and in inequality form) Interval: Inequality:

Now we have enough information to describe this piecewise function:

for _______________________( )

________ for ________ ____ __ _f x

=

What this fancy equation tells us is that there are two parts to our function – and where these two parts are “valid.” Evaluate, using the graph:

( 2)f − =

( 1)f − =

(0)f =

(1)f =

(2)f =

Evaluate, using the equation:

( 2)f − =

( 1)f − =

(0)f =

(1)f =

(2)f =

−4 −3 −2 −1 1 2 3 4 5

−2

−1

1

2

3

4

5

6

7

x

y

2

Now that we’ve talked about piecewise functions, try this on your own:

Write the piecewise function:

for _______________________( )

________ for ________ ____ __ _g x

=

Evaluate the piecewise function at the following x-values:

( 6)g − =

(1)g =

(2)g =

Let’s do things a little differently now. I’m going to give you the piecewise function, and I want you to use it to evaluate it at some x-values, and then graph it.

2

1for 3

3

Kai for -3 1

1

(

fo

)

1

2

r

x

x

x

x

x

x

−

=

≤ −

< ≤

>

Kai( 6)− = Kai( 2.999)− =

Kai( 3)− = Kai( 3.001)− =

Kai(0) = Kai(1.8) =

Kai(1) = Kai(10) =

How many pieces are there in this piecewise function? __________ When you graph the piecewise function, what will the left-most piece look like? When you graph the piecewise function , what will the middle piece look like? When you graph the piecewise function, what will the right-most piece look like?

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

−8

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

7

8

x

y

3

Now to graph!

2

1for 3

3

Kai for -3 1

1

(

fo

)

1

2

r

x

x

x

x

x

x

−

=

≤ −

< ≤

>

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−9

−8

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

7

8

9

x

y

4

Problems!

Evaluate

( 6)

( 5)

(21)

f

f

f

− =

− =

=

1. Let

2. Let 2

for 2

for

9(

2

5)

4

xg x

x

x

x

<

≥

−=

− Evaluate

( 6)

(3)

(2)

g

g

g

− =

=

=

3-5: Write the piecewise function for the following graphs: 3. Graph

Equation of the pieces Domain for the pieces Piecewise Function

4. Graph

Equation of the pieces Domain for the pieces Piecewise Function

5. Graph

Equation of the pieces Domain for the pieces Piecewise Function

8

( ) 0

for 5

fo 5

o5

r 5

f r 5

x

f xx

x

−

= − < <

≥

≤

5

6. This is the mondo-mega problem. Graph the following piecewise function:

3

2

1

| |

for

for 1 2Ian( )

for 3

for 3

2

5

xx

x x

x

x

x

x

≤ <=

≤ <

< −

−

− ≥

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10 11

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

7

8

9

10

x

y

6