Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Domain and Range of a Transformation, Study Guides, Projects, Research of Signals and Systems

When a function is transformed, its domain and/or range will change. If only the inputs are transformed, then only the domain will change. If only.

Typology: Study Guides, Projects, Research

2021/2022

Uploaded on 09/12/2022

parvini
parvini ๐Ÿ‡บ๐Ÿ‡ธ

4.5

(15)

243 documents

1 / 5

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
16-week Lesson 22 (8-week Lesson 18) Domain and Range of a Transformation
1
When a function is transformed, its domain and/or range will change. If
only the inputs are transformed, then only the domain will change. If only
the outputs are transformed, then only the range will change. If both the
inputs and outputs are transformed, then both the domain and range will
change.
Remember that the domain represents the set of inputs for a function,
and the range represents the set of outputs.
Example 1: Let ๐‘ฆ=๐‘“(๐‘ฅ) be a function with domain ๐ท = [โˆ’6,5] and
range ๐‘…=[0,14]. Find the domain ๐ท and range ๐‘… for each of the
following functions. Keep in mind order of operation and the order of
your intervals.
a. ๐‘ฆ=โˆ’3๐‘“(๐‘ฅ) b. ๐‘ฆ=๐‘“(1
2๐‘ฅ)
Changes INside the parentheses change the INputs and we do the
INverse; remember that the Domain is the set of inputs
Changes OUTside the parentheses change the OUTputs and we do
exactly what we see; remember that the Range is the set of outputs
inputs,โˆ’3(outputs) inputs
1
2,outputs
Range:[โˆ’3(0),โˆ’3(14)] Domain:[โˆ’6
1
2,5
1
2]
Range:[0,โˆ’42] Domain:[โˆ’12,10]
๐ƒ๐จ๐ฆ๐š๐ข๐ง:[โˆ’๐Ÿ”,๐Ÿ“] ๐ƒ๐จ๐ฆ๐š๐ข๐ง:[โˆ’๐Ÿ๐Ÿ,๐Ÿ๐ŸŽ]
๐‘๐š๐ง๐ ๐ž:[โˆ’๐Ÿ’๐Ÿ,๐ŸŽ] ๐‘๐š๐ง๐ ๐ž:[๐ŸŽ,๐Ÿ๐Ÿ’]
Since the inputs of this function are not
being changed with the transformation
๐‘ฆ= โˆ’3๐‘“(๐‘ฅ), that means the domain is
also not being changed. So the domain
will still be ๐ท=[โˆ’6,5].
Since the outputs of this function are
not being changed with the
transformation ๐‘ฆ=๐‘“(1
2๐‘ฅ), that means
the range is also not being changed. So
the range will remain ๐ท=[0,14].
pf3
pf4
pf5

Partial preview of the text

Download Domain and Range of a Transformation and more Study Guides, Projects, Research Signals and Systems in PDF only on Docsity!

When a function is transformed, its domain and/or range will change. If

only the inputs are transformed, then only the domain will change. If only

the outputs are transformed, then only the range will change. If both the

inputs and outputs are transformed, then both the domain and range will

change.

Remember that the domain represents the set of inputs for a function,

and the range represents the set of outputs.

Example 1 : Let ๐‘ฆ = ๐‘“

be a function with domain ๐ท =

[

]

and

range ๐‘… =

[

]

. Find the domain ๐ท and range ๐‘… for each of the

following functions. Keep in mind order of operation and the order of

your intervals.

a. ๐‘ฆ = โˆ’ 3 ๐‘“

b. ๐‘ฆ = ๐‘“ (

1

2

Changes IN side the parentheses change the IN puts and we do the

IN verse; remember that the Domain is the set of inputs

Changes OUT side the parentheses change the OUT puts and we do

exactly what we see; remember that the Range is the set of outputs

inputs, โˆ’ 3

outputs

inputs

1

2

, outputs

Range:

[

)]

Domain: [โˆ’

6

1

2

5

1

2

]

Range:

[

]

Domain:

[

]

[

]

[

]

[

]

[

]

Since the inputs of this function are not

being changed with the transformation

๐‘ฆ = โˆ’ 3 ๐‘“

( ๐‘ฅ

) , that means the domain is

also not being changed. So the domain

will still be ๐ท = [โˆ’ 6 , 5 ].

Since the outputs of this function are

not being changed with the

transformation ๐‘ฆ = ๐‘“ (

1

2

๐‘ฅ), that means

the range is also not being changed. So

the range will remain ๐ท = [ 0 , 14 ].

Be sure to keep in mind that intervals (such as a domain or range),

just like number lines, always go in order from smallest to largest as

you go from left to right. On Example 1a, the range is listed as

[

]

because โˆ’๐Ÿ’๐Ÿ is smaller than ๐ŸŽ.

Be sure to re-arrange intervals as needed so they are in the correct

order.

Example 2 : Let ๐‘ฆ = ๐‘“(๐‘ฅ) be a function with domain ๐ท =

[

]

and

range ๐‘… = [ 0 , 14 ]. Find the domain ๐ท and range ๐‘… for each of the

following functions. Keep in mind order of operation and the order of

your intervals.

a. ๐‘ฆ = ๐‘“

โˆ’ 2 b. ๐‘ฆ = ๐‘“

b.

Example 3 : Let ๐‘ฆ = ๐‘“

be a function with domain ๐ท =

[

]

and

range ๐‘… =

[

]

. Find the domain ๐ท and range ๐‘… for each of the

following functions. Keep in mind order of operation and the order of

your intervals.

a. ๐‘ฆ =

1

2

b. ๐‘ฆ = โˆ’๐‘“

b.

Example 6 : Let ๐‘ฆ = ๐‘“

be a function with domain ๐ท =

[

and

range ๐‘… =

]

. Find the domain ๐ท and range ๐‘… for each of the

following functions. Keep in mind order of operation and the order of

your intervals.

a. ๐‘ฆ =

2

3

โˆ’ 1 b. ๐‘ฆ = โˆ’ 3 ๐‘“ (โˆ’

1

3

Example 7 : Let ๐‘ฆ = ๐‘“(๐‘ฅ) be a function with domain ๐ท = [โˆ’ 9 , 0 ] and

range ๐‘… =

. Find the domain ๐ท and range ๐‘… for each of the

following functions. Keep in mind order of operation and the order of

your intervals.

a. ๐‘ฆ = ๐‘“

  • 1 b. ๐‘ฆ = โˆ’๐‘“

b. a

inputs + 4 , outputs + 1

inputs

3

outputs

Domain:

[

]

Domain: [โˆ’

9

3

0

3

]

Range:

Range: (โˆ’ 1

Domain:

[

]

Domain:

[

]

Range:

Range:

[

]

[

]

Example 8 : Let ๐‘ฆ = ๐‘“

be a function with domain ๐ท =

and

range ๐‘… =

[

]

. Find the domain ๐ท and range ๐‘… for each of the

following functions. Keep in mind order of operation and the order of

your intervals.

a. ๐‘ฆ =

1

2

  • 3 b. ๐‘ฆ = โˆ’ 5 ๐‘“

Once again, keep in mind that the domain of a function is the set of inputs,

while the range of a function is the set of outputs. So any changes to the

inputs of a function are made to the domain, and any changes to the

outputs of a function are made to the range.

Answers to Examples:

1 a. ๐ท:

[

]

[

]

; 1 b. ๐ท:

[

]

[

]

2a. ๐ท:

[

]

[

]

; 2b. :

[

]

[

]

3a. :

[

]

[

]

; 3b. ๐ท: [โˆ’ 2 ,

5

3

] , ๐‘…:

[

]

4a. :

[

]

, ๐‘…: [โˆ’ 1 ,

25

3

] ; 4b. ๐ท: [โˆ’

10

3

, 4 ] , ๐‘…:

[

]

5 a. ๐ท: (โˆ’โˆž, 0 ], ๐‘…: (โˆ’โˆž, 3 ] ; 5 b. ๐ท: [ 0 , โˆž), ๐‘…: [โˆ’ 2 , โˆž) ;

6a. ๐ท:

[

]

; 6b. ๐ท:

]

[

7 a. ๐ท:

[

]

; 7 b. ๐ท:

[

]

8 a. ๐ท: (โˆ’โˆž, โˆž), ๐‘…: [

1

2

, 5 ] ; 8 b. ๐ท: (โˆ’โˆž, โˆž), ๐‘…: [โˆ’ 22 , 23 ] ;