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Graphing Piecewise Defined Functions: Creating Input/Output Tables and Sketching Graphs, Schemes and Mind Maps of Calculus

Instructions on graphing piecewise defined functions by creating input/output tables for each piece and plotting the ordered pairs. examples of piecewise functions and their corresponding graphs, as well as instructions for setting up tables in LON-CAPA. Students are encouraged to find a minimum of three points for each piece of the function, including endpoints.

What you will learn

  • How do you determine the domain and range of a piecewise defined function?
  • How do you create input/output tables for piecewise defined functions?
  • What is the importance of finding endpoints when graphing piecewise defined functions?

Typology: Schemes and Mind Maps

2021/2022

Uploaded on 09/27/2022

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16-week Lesson 25 (8-week Lesson 20) Graphing Piecewise Defined Functions
1
Piecewise-defined function:
- function described by more than one expression
o ๐‘“(๐‘ฅ)= { โˆ’7 if ๐‘ฅ<โˆ’3
๐‘ฅ2 if โˆ’3โ‰ค๐‘ฅ<2
๐‘ฅโˆ’8 if ๐‘ฅโ‰ฅ2
- keep in mind a piecewise-defined function must meet the
requirements of a function; this means that every input from the
domain has only one output
o when graphing, there will be some inputs that get plugged into
to more than one piece of the function (such as โˆ’3 and 2 in the
function given above), but only one of the points that results will
be part of the graph; points that are NOT part of the graph are
denoted as empty points ( )
- when graphing piecewise defined functions, I will use input/output
tables
o I will create one input/output table for each piece of the
piecewise function
o the homework problems are set-up in the same way (complete a
table to find order pairs, then plot the ordered pairs)
For the piecewise function given above, I would have the following tables
(note that each domain below is listed using interval notation rather than
inequalities; this is to emphasize when a domain goes to โˆ’โˆž or โˆž):
I would then plot these points to graph the piecewise function ๐‘“.
๐’‡(๐’™)=โˆ’๐Ÿ•
(โˆ’โˆž,โˆ’๐Ÿ‘)
Inputs
Outputs
๐‘ฅ
๐‘“(๐‘ฅ)
โˆ’3
โˆ’7
โˆ’4
โˆ’7
โˆ’5
โˆ’7
Inputs
Outputs
๐‘ฅ
๐‘“(๐‘ฅ)
โˆ’3
9
โˆ’2
4
โˆ’1
1
0
0
1
1
2
4
๐’‡(๐’™)=๐’™โˆ’๐Ÿ–
[๐Ÿ,โˆž)
Inputs
Outputs
๐‘ฅ
๐‘“(๐‘ฅ)
2
โˆ’6
3
โˆ’5
4
โˆ’4
pf3
pf4
pf5

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Piecewise-defined function:

  • function described by more than one expression

o ๐‘“

โˆ’ 7 if ๐‘ฅ < โˆ’ 3

2

if โˆ’ 3 โ‰ค ๐‘ฅ < 2

๐‘ฅ โˆ’ 8 if ๐‘ฅ โ‰ฅ 2

  • keep in mind a piecewise-defined function must meet the

requirements of a function; this means that every input from the

domain has only one output

o when graphing, there will be some inputs that get plugged into

to more than one piece of the function (such as โˆ’ 3 and 2 in the

function given above), but only one of the points that results will

be part of the graph; points that are NOT part of the graph are

denoted as empty points ( )

  • when graphing piecewise defined functions, I will use input/output

tables

o I will create one input/output table for each piece of the

piecewise function

o the homework problems are set-up in the same way (complete a

table to find order pairs, then plot the ordered pairs)

For the piecewise function given above, I would have the following tables

(note that each domain below is listed using interval notation rather than

inequalities; this is to emphasize when a domain goes to โˆ’โˆž or โˆž):

I would then plot these points to graph the piecewise function ๐‘“.

Inputs Outputs

๐Ÿ

[โˆ’๐Ÿ‘, ๐Ÿ)

Inputs Outputs

[

Inputs Outputs

Example 1 : Sketch the graph of the piecewise-defined function ๐‘”.

2 ๐‘ฅ + 3 if ๐‘ฅ โ‰ค โˆ’ 2

2

if โˆ’ 2 < ๐‘ฅ โ‰ค 2

๐‘ฅ โˆ’ 2 if ๐‘ฅ > 2

Once again, keep in mind that the point

is not part of the graph

because the input 0 is not part of the domain of the first piece. Now I will

make another input/output table for the second piece, and then sketch its

]

Inputs Outputs

๐Ÿ

โˆ’๐Ÿ < ๐’™ โ‰ค ๐Ÿ ๐จ๐ซ (โˆ’๐Ÿ, ๐Ÿ]

Inputs Outputs

Inputs Outputs

Since the first piece

of the function

( 2 ๐‘ฅ + 3 ) is only

defined for ๐‘ฅ-values

where ๐‘ฅ โ‰ค โˆ’ 2 , its

domain in interval

notation is

(โˆ’โˆž, โˆ’ 2 ]. This

means that the

graph of the first

piece will go on

forever in the

negative direction.

The last point that I

plotted for the

graph of the first

piece is (โˆ’ 4 , โˆ’ 5 ),

but the graph

continues on past

that point since I

could continue to

plug in other ๐‘ฅ-

values (such as

โˆ’ 5 , โˆ’ 6 , โˆ’ 7 , โ€ฆ).

The graph of the

third piece will also

go on forever, but

in the positive

direction, since its

domain is

( 2 , โˆž

) .

Example 2 : Sketch the graph of the piecewise-defined function โ„Ž.

โˆš 1 โˆ’ ๐‘ฅ if ๐‘ฅ โ‰ค 1

2

โˆ’ 10 if 1 < ๐‘ฅ < 4

โˆ’๐‘ฅ if ๐‘ฅ โ‰ฅ 4

(โˆ’โˆž, ๐Ÿ]

Inputs Outputs

๐Ÿ

Inputs Outputs

[

Inputs Outputs

Example 3 : Sketch the graph of the piecewise-defined function ๐‘—.

2

if ๐‘ฅ โ‰ค โˆ’ 2

2 ๐‘ฅ + 2 if โˆ’ 2 < ๐‘ฅ โ‰ค 2

2

+ 5 if ๐‘ฅ > 2

๐Ÿ

]

Inputs Outputs

(โˆ’๐Ÿ, ๐Ÿ]

Inputs Outputs

๐Ÿ

Inputs Outputs

Keep in mind that

even though the

second piece of the

function ( 2 ๐‘ฅ + 2 )

has the domain

โˆ’ 2 < ๐‘ฅ โ‰ค 2 (which

means that โˆ’ 2 is not

part of the domain)

we still need to plug

โˆ’ 2 into 2 ๐‘ฅ + 2 in

order to get the

complete graph of

that piece of the

function. If we

started graphing that

piece of the function

using ๐‘ฅ = โˆ’ 1 as the

first input, weโ€™d be

missing part of the

graph, and the graph

would be incomplete.

So we include

๐‘ฅ = โˆ’ 2 as an input

for the second piece

of the function, but

we plot that point

with an open dot to

indicate that ๐‘ฅ = โˆ’ 2

is not part of the

domain of that piece,

and that the point

(โˆ’ 2 , โˆ’ 2 ) is not part

of the graph.