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 푦 = 푓(푥) be a function with domain 퐷 = [−6, 5] and range 푅 = [0, 14]. Find the domain 퐷 and range 푅 for each of the following functions. Keep in mind order of operation and the order of your intervals. 1 a. 푦 = −3푓(푥) b. 푦 = 푓 ( 푥) 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 , outputs) 2

Since the inputs of this function are not Since the outputs of this function are being changed with the transformation not being changed with the 1 푦 = −3푓(푥), that means the domain is transformation 푦 = 푓 ( 푥), that means also not being changed. So the domain 2 the range is also not being changed. So will still be 퐷 = [−6, 5]. the range will remain 퐷 = [0, 14].

6 5 Range: [−3(0), −3(14)] Domain: [− 1 , 1 ] 2 2

Range: [0, −42] Domain: [−12, 10]

퐃퐨퐦퐚퐢퐧: [−ퟔ, ퟓ] 퐃퐨퐦퐚퐢퐧: [−ퟏퟐ, ퟏퟎ] 퐑퐚퐧퐠퐞: [−ퟒퟐ, ퟎ] 퐑퐚퐧퐠퐞: [ퟎ, ퟏퟒ] 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 [−ퟒퟐ, ퟎ] because −ퟒퟐ is smaller than ퟎ. Be sure to re-arrange intervals as needed so they are in the correct order.

Also, remember that a domain is a set of inputs and a range is a set of outputs.

Example 2: Let 푦 = 푓(푥) be a function with domain 퐷 = [−6, 5] and range 푅 = [0, 14]. Find the domain 퐷 and range 푅 for each of the following functions. Keep in mind order of operation and the order of your intervals.

a. 푦 = 푓(푥 + 3) − 2 b. 푦 = 푓(푥 − 4) + 1

b. Example 3: Let 푦 = 푓(푥) be a function with domain 퐷 = [−6, 5] and range 푅 = [0, 14]. Find the domain 퐷 and range 푅 for each of the following functions. Keep in mind order of operation and the order of your intervals.

1 a. 푦 = 푓(−푥) b. 푦 = −푓(3푥) 2 b.

2

16-week Lesson 22 (8-week Lesson 18) Domain and Range of a Transformation Example 4: Let 푦 = 푓(푥) be a function with domain 퐷 = [−6, 5] and range 푅 = [0, 14]. Find the domain 퐷 and range 푅 for each of the following functions. Keep in mind order of operation and the order of your intervals.

2 3 a. 푦 = 푓(푥) − 1 b. 푦 = −푓 (− 푥) 3 2

Example 5: Let 푦 = 푓(푥) be a function with domain 퐷 = [0, ∞) and range 푅 = (−∞, 0]. Find the domain 퐷 and range 푅 for each of the following functions. Keep in mind order of operation and the order of your intervals. 1 a. 푦 = 푓(−푥) + 3 b. 푦 = −푓(2푥) − 2 2

b. 2 1 c. 푦 = 푓(푥 − 4) − 1 d. 푦 = −3푓 (− 푥) 3 3

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16-week Lesson 22 (8-week Lesson 18) Domain and Range of a Transformation Example 6: Let 푦 = 푓(푥) be a function with domain 퐷 = [−9, 0] and range 푅 = (−∞, ∞). Find the domain 퐷 and range 푅 for each of the following functions. Keep in mind order of operation and the order of your intervals. a. 푦 = 푓(푥 − 4) + 1 b. 푦 = −푓(3푥) b. a

Example 7: Let 푦 = 푓(푥) be a function with domain 퐷 = (−∞, ∞) and range 푅 = [−5, 4]. Find the domain 퐷 and range 푅 for each of the following functions. Keep in mind order of operation and the order of your intervals. 1 a. 푦 = 푓(−푥) + 3 b. 푦 = −5푓(2푥) − 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. 퐷: [−6, 5], 푅: [−42, 0] ; 1b. 퐷: [−12, 10], 푅: [0, 14] ; 2a. 퐷: [−9, 2], 푅: [−2, 12] ; 2b. : [−2, 9], 푅: [1, 15] ; 5 3a. : [−5, 6], 푅: [0, 7] ; 3b. 퐷: [−2, ] , 푅: [−14, 0] ; 3 25 10 4a. : [−6, 5], 푅: [−1, ] ; 4b. 퐷: [− , 4] , 푅: [−14, 0] ; 3 3 5a. 퐷: (−∞, 0], 푅: (−∞, 3] ; 5b. 퐷: [0, ∞), 푅: [−2, ∞) ; 5c. 퐷: [4, ∞), 푅: (−∞, −1] ; 5d. 퐷: (−∞, 0], 푅: [0, ∞) ; 6a. 퐷: [−5, 4], 푅: (−∞, ∞) ; 6b. 퐷: [−3, 0], 푅: (−∞, ∞) ; 1 7a. 퐷: (−∞, ∞), 푅: [ , 5] ; 7b. 퐷: (−∞, ∞), 푅: [−22, 23] ; 2 4