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

University of North Texas Health Science Center (UNTHSC)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.

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 𝑓(𝑥)= { −7 if 𝑥<−3

𝑥2 if −3≤𝑥<2

𝑥−8 if 𝑥≥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 −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 −∞ or ∞):

I would then plot these points to graph the piecewise function 𝑓.

𝒇(𝒙)=−𝟕

(−∞,−𝟑)

Inputs

Outputs

𝑥

𝑓(𝑥)

−3

−7

−4

−7

−5

−7

𝒇(𝒙)=𝒙𝟐

[−𝟑,𝟐)

Inputs

Outputs

𝑥

𝑓(𝑥)

−3

9

−2

4

−1

1

0

1

2

4

𝒇(𝒙)=𝒙−𝟖

[𝟐,∞)

Inputs

Outputs

𝑥

𝑓(𝑥)

2

−6

3

−5

4

−4

Discover Schemes and Mind Maps of Calculus University of North Texas Health Science Center (UNTHSC)

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Piecewise-defined function:

function described by more than one expression

o 𝑓

− 7 if 𝑥 < − 3

2

if − 3 ≤ 𝑥 < 2

𝑥 − 8 if 𝑥 ≥ 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 − 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 −∞ or ∞):

I would then plot these points to graph the piecewise function 𝑓.

Inputs Outputs

𝟐

[−𝟑, 𝟐)

Inputs Outputs

[

Inputs Outputs

Example 1 : Sketch the graph of the piecewise-defined function 𝑔.

2 𝑥 + 3 if 𝑥 ≤ − 2

2

if − 2 < 𝑥 ≤ 2

𝑥 − 2 if 𝑥 > 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

𝟐

−𝟐 < 𝒙 ≤ 𝟐 𝐨𝐫 (−𝟐, 𝟐]

Inputs Outputs

Since the first piece

of the function

( 2 𝑥 + 3 ) is only

defined for 𝑥-values

where 𝑥 ≤ − 2 , its

domain in interval

notation is

(−∞, − 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 (− 4 , − 5 ),

but the graph

continues on past

that point since I

could continue to

plug in other 𝑥-

values (such as

− 5 , − 6 , − 7 , …).

The graph of the

third piece will also

go on forever, but

in the positive

direction, since its

domain is

( 2 , ∞

) .

Example 2 : Sketch the graph of the piecewise-defined function ℎ.

√ 1 − 𝑥 if 𝑥 ≤ 1

2

− 10 if 1 < 𝑥 < 4

−𝑥 if 𝑥 ≥ 4

(−∞, 𝟏]

Inputs Outputs

𝟐

Inputs Outputs

[

Inputs Outputs

Example 3 : Sketch the graph of the piecewise-defined function 𝑗.

2

if 𝑥 ≤ − 2

2 𝑥 + 2 if − 2 < 𝑥 ≤ 2

2

+ 5 if 𝑥 > 2

𝟐

]

Inputs Outputs

(−𝟐, 𝟐]

Inputs Outputs

𝟐

Inputs Outputs

Keep in mind that

even though the

second piece of the

function ( 2 𝑥 + 2 )

has the domain

− 2 < 𝑥 ≤ 2 (which

means that − 2 is not

part of the domain)

we still need to plug

− 2 into 2 𝑥 + 2 in

order to get the

complete graph of

that piece of the

function. If we

started graphing that

piece of the function

using 𝑥 = − 1 as the

first input, we’d be

missing part of the

graph, and the graph

would be incomplete.

So we include

𝑥 = − 2 as an input

for the second piece

of the function, but

we plot that point

with an open dot to

indicate that 𝑥 = − 2

is not part of the

domain of that piece,

and that the point

(− 2 , − 2 ) is not part

of the graph.

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

Related documents

Partial preview of the text

Download Graphing Piecewise Defined Functions: Creating Input/Output Tables and Sketching Graphs and more Schemes and Mind Maps Calculus in PDF only on Docsity!

Piecewise-defined function:

o 𝑓

− 7 if 𝑥 < − 3

if − 3 ≤ 𝑥 < 2

𝑥 − 8 if 𝑥 ≥ 2

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 − 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 ( )

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 −∞ or ∞):

I would then plot these points to graph the piecewise function 𝑓.

[−𝟑, 𝟐)

[

Example 1 : Sketch the graph of the piecewise-defined function 𝑔.

2 𝑥 + 3 if 𝑥 ≤ − 2

if − 2 < 𝑥 ≤ 2

𝑥 − 2 if 𝑥 > 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

]

−𝟐 < 𝒙 ≤ 𝟐 𝐨𝐫 (−𝟐, 𝟐]

Example 2 : Sketch the graph of the piecewise-defined function ℎ.

√ 1 − 𝑥 if 𝑥 ≤ 1

− 10 if 1 < 𝑥 < 4

−𝑥 if 𝑥 ≥ 4

(−∞, 𝟏]

[

Example 3 : Sketch the graph of the piecewise-defined function 𝑗.

if 𝑥 ≤ − 2

2 𝑥 + 2 if − 2 < 𝑥 ≤ 2

+ 5 if 𝑥 > 2

]

(−𝟐, 𝟐]