Domain and Range of a Transformation - Purdue University

16-week Lesson 22 (8-week Lesson 18)

Domain and Range of a Transformation

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 ? = ?(? ) be a function with domain ? = [?6, 5] and

range ? = [0, 14]. Find the domain ? and range ? for each of the

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

your intervals.

1

a. ? = ?3?(? )

b. ? = ? (2 ?)

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

inputs, ?3(outputs)

1

2

, outputs

Since the outputs of this function are

not being changed with the

1

transformation ? = ? ( ?), that means

2

the range is also not being changed. So

the range will remain ? = [0, 14].

Since the inputs of this function are not

being changed with the transformation

? = ?3?(?), that means the domain is

also not being changed. So the domain

will still be ? = [?6, 5].

Range: [?3(0), ?3(14)]

6

5

2

2

Domain: [? 1 , 1 ]

Range: [0, ?42]

Domain: [?12, 10]

??????: [??, ?]

?????: [???, ?]

??????: [???, ??]

?????: [?, ??]

1

16-week Lesson 22 (8-week Lesson 18)

Domain and Range of a Transformation

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 ??? is smaller than ?.

?��

��

?=0

? = ?42

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

order.

Example 2: Let ? = ?(? ) be a function with domain ? = [?6, 5] and

range ? = [0, 14]. Find the domain ? and range ? for each of the

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

your intervals.

a. ? = ?(? + 3) ? 2

b. ? = ? (? ? 4) + 1

b.

Example 3: Let ? = ?(? ) be a function with domain ? = [?6, 5] and

range ? = [0, 14]. Find the domain ? and range ? for each of the

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

your intervals.

1

a. ? = 2 ?(?? )

b.

b. ? = ??(3? )

2

16-week Lesson 22 (8-week Lesson 18)

Domain and Range of a Transformation

Example 4: Let ? = ?(? ) be a function with domain ? = [?6, 5] and

range ? = [0, 14]. Find the domain ? and range ? for each of the

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

your intervals.

2

3

a. ? = 3 ?(? ) ? 1

b. ? = ?? (? 2 ?)

Example 5: Let ? = ?(? ) be a function with domain ? = [0, ��) and

range ? = (?��, 0]. Find the domain ? and range ? for each of the

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

your intervals.

1

a. ? = 2 ?(?? ) + 3

b. ? = ??(2? ) ? 2

inputs 1

?1

inputs

, 2 (outputs) + 3

0

2

��

Domain: [?1 , ?1)

1

, ?1(outputs) ? 2

0 ��

Domain: [2 , 2 )

1

Range: (2 (?��) + 3, 2 (0) + 3]

Range: (?1(?��) ? 2, ?1(0) ? 2]

Domain: [0, ?��)

Range: (?��, 3]

Domain: [0, ��)

Range: (��, ?2]

??????: (?��, ?]

?????: (?��, ?]

??????: [?, ��)

?????: [??, ��)

3

16-week Lesson 22 (8-week Lesson 18)

Domain and Range of a Transformation

Example 6: Let ? = ?(? ) be a function with domain ? = [0, ��) and

range ? = (?��, 0]. Find the domain ? and range ? for each of the

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

your intervals.

2

1

a. ? = 3 ?(? ? 4) ? 1

b. ? = ?3? (? 3 ?)

Example 7: Let ? = ?(? ) be a function with domain ? = [?9, 0] and

range ? = (?��, ��). Find the domain ? and range ? for each of the

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

your intervals.

a. ? = ?(? ? 4) + 1

b. ? = ??(3? )

b. a

inputs

inputs + 4, outputs + 1

, ?1(outputs)

3

9 0

Domain: [?9 + 4, 0 + 4]

Domain: [? 3 , 3]

Range: (?�� + 1, �� + 1)

Range: (?1(?��), ?1(��))

Domain: [?5, 4]

Range: (?��, ��)

Domain: [?3, 0]

Range: (��, ?��)

??????: [??, ?]

?????(?��, ��)

??????: [??, ?]

?????: (?��, ��)

4

16-week Lesson 22 (8-week Lesson 18)

Domain and Range of a Transformation

Example 8: Let ? = ?(? ) be a function with domain ? = (?��, ��) and

range ? = [?5, 4]. Find the domain ? and range ? for each of the

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

your intervals.

1

a. ? = 2 ?(?? ) + 3

b. ? = ?5? (2? ) ? 2

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:

1a. ?: [?6, 5], ?: [?42, 0] ; 1b. ?: [?12, 10], ?: [0, 14] ;

2a. ?: [?9, 2], ?: [?2, 12] ; 2b. : [?2, 9], ?: [1, 15] ;

5

3a. : [?5, 6], ?: [0, 7] ; 3b. ?: [?2, 3] , ?: [?14, 0] ;

25

10

4a. : [?6, 5], ?: [?1, 3 ] ; 4b. ?: [? 3 , 4] , ?: [?14, 0] ;

5a. ?: (?��, 0], ?: (?��, 3] ; 5b. ?: [0, ��), ?: [?2, ��) ;

6a. ?: [4, ��), ?: (?��, ?1] ; 6b. ?: (?��, 0], ?: [0, ��) ;

7a. ?: [?5, 4], ?: (?��, ��) ; 7b. ?: [?3, 0], ?: (?��, ��) ;

1

8a. ?: (?��, ��), ?: [ , 5] ; 8b. ?: (?��, ��), ?: [?22, 23] ;

2

5

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