Math 3 Unit 6: Radical Functions

Unit Title Standards

6.1 Simplifying Radical Expressions N.RN.2, A.SSE.2

6.2 Multiplying and Dividing Radical Expressions N.RN.2, F.IF.8

6.3 Adding & Subtracting Radical Expressions N.RN.2, A.SSE.2

6.4 Multiplying & Dividing Binomial Radical Expressions N.RN.2, A.SSE.2

6.5 Rational Exponents N.RN.1, N.RN.2

6.6 Solving Radical Equations A.REI.2

6.7 Graphing Radical Equations F.IF.7B, F.IF.5

6.8 Graphing Radical Equations with Cubed Roots F.IF.7B, F.IF.5

6.9 Solving and Graphing Radical Equations A.REI.11

Unit 6 Review

Additional Clovis Unified Resources http://mathhelp.cusd.com/courses/math-3 Clovis Unified is dedicated to helping you be successful in Math 3. On the website above you will find videos from Clovis Unified teachers on lessons, homework, and reviews. Digital copies of the worksheets, as well as hyperlinks to the videos listed on the back are also available at this site.

Math 3 Unit 6: Online Resources 6.1 Simplifying

Radical Expressions

• Khan Academy: Simplifying Radical Terms http://bit.ly/61srea or http://bit.ly/61sreb

• Purple Math: Square Roots & More Simplification http://bit.ly/61srec or http://bit.ly/61sree

• Patrick JMT: Radical Notation and Simplifying Radicals (Basic) http://bit.ly/61sref

6.2 Multiplying and Dividing Radical Expressions

• Khan Academy: Rationalizing the Denominator http://bit.ly/62mdrea

• Khan Academy: Rational Expressions - Multiplying and Dividing http://bit.ly/62mdreb or http://bit.ly/62mdrec

• Purple Math: Rationalizing Denominators http://bit.ly/62mdred

6.3 Adding & Subtracting Radical Expressions

• Khan Academy: Adding Radical Expressions http://bit.ly/63asreb

• Khan Academy: Subtracting Radical Expressions http://bit.ly/63asrec

• Purple Math: Adding and Subtracting Radical Expressions http://bit.ly/63asred

6.4 Multiplying & Dividing Binomial Radical Expressions

• Open Algebra: Multiplying and Dividing Radical Expressions (multiple video links at the end) http://bit.ly/64mdba

• Multiplying Binomial Radical Expressions http://bit.ly/64mdbb

6.5 Rational Exponents

• Patrick JMT: Evaluating Numbers with Rational Exponents by using Radical Notation http://bit.ly/65raexa

• Patrick JMT: Multiplying Variables with Rational Exponents – Basic Ex 1 & Ex 2 http://bit.ly/65raexb or http://bit.ly/65raexc

• Khan Academy: Simplifying Rational Exponents http://bit.ly/65raexd

6.6 Solving Radical Equations

• Patrick JMT: Solving Equations Involving Rational Exponents http://bit.ly/66srea

• Patrick JMT: Solving Equations Involving Square Roots http://bit.ly/66sreb

• Khan Academy: Solving Square-Root Equations http://bit.ly/66srec

6.7 Graphing Radical Equations

• Khan Academy: Graphs of Square-Root Functions http://bit.ly/67grea

• Khan Academy: Square-Root Functions & their Graphs http://bit.ly/67greb

6.8 Graphing Radical Equations with Cubed Roots

• Math Bits Notebook: Square-Root & Cube Root Functions http://bit.ly/68grea

• Snap Math: Domain and Range of Cubed Root http://bit.ly/68greb

• Graphing Cube Root Functions http://bit.ly/68grec

6.9 Solving and Graphing Radical Equations

• Purple Math: Graphing Radical Functions (Pages 1-3) http://bit.ly/69sgrea

• Patrick JMT: Solving an Equation Involving a Single Radical (Square Root) – Ex 1 http://bit.ly/69sgreb

• Patrick JMT: Solving an Equation Containing Two Radicals – Ex 1 http://bit.ly/69sgrec

Math 3 Unit 6 Worksheet 1

Math 3 Unit 6 Worksheet 1 Name: Simplifying Radical Expressions Date: Per:

[1-6] Simplify.

1. √25 2. √0.09 3. � 49121

4. √273 5. √– 83 6. � 64125

3

[7- 9] Find each real root.

7. √36 8. −√273 9. √0.01

[10-29] Simplify each radical expression. Use absolute value symbols when needed.

10. √25𝑥𝑥2 11. √𝑎𝑎6𝑏𝑏12 12. √9𝑐𝑐4𝑑𝑑2 13. �16𝑥𝑥2𝑦𝑦8

14. �121𝑥𝑥3𝑦𝑦12 15. −3𝑥𝑥�81𝑥𝑥7𝑦𝑦2 16. 2√24𝑎𝑎4𝑏𝑏9 17. 𝑎𝑎√27𝑎𝑎5𝑏𝑏6

18. 3𝑥𝑥2�40𝑥𝑥𝑦𝑦4 19. √75𝑎𝑎8𝑏𝑏7 20. √8𝑥𝑥33 21. �27𝑥𝑥12𝑦𝑦153

Math 3 Unit 6 Worksheet 1

22. �−64𝑥𝑥3𝑦𝑦63 23. �𝑥𝑥14𝑦𝑦53 24. √−𝑎𝑎𝑏𝑏3𝑐𝑐43 25. 2𝑥𝑥�−24𝑥𝑥3𝑦𝑦53

26. �32𝑥𝑥9𝑦𝑦103 27. �16𝑥𝑥4𝑦𝑦93 28. 3𝑥𝑥2 √−40𝑥𝑥83 29. √−54𝑎𝑎6𝑏𝑏11𝑐𝑐73

[30-34] Find all the real solutions of each equation.

30. 𝑥𝑥2 = 36 31. 𝑥𝑥3 = 8 32. 𝑥𝑥2 = 0.81 33. 𝑥𝑥3 = −27 34. 𝑥𝑥2 = 16

35. Determine whether each expression is equivalent to �32𝑥𝑥3𝑦𝑦2. Select Yes or No for each expression.

36. Determine whether each expression is equivalent to �64𝑥𝑥6𝑦𝑦53 . Select Yes or No for each expression.

Yes No 2𝑥𝑥�8𝑥𝑥𝑦𝑦2

4𝑥𝑥𝑦𝑦√2𝑥𝑥

4𝑥𝑥|𝑦𝑦|√2𝑥𝑥

16𝑥𝑥�𝑥𝑥𝑦𝑦2

Yes No 2|𝑥𝑥| �8𝑥𝑥𝑦𝑦53

4𝑥𝑥 �𝑥𝑥𝑦𝑦53

4𝑥𝑥2𝑦𝑦2 �𝑦𝑦33

(2𝑥𝑥)2𝑦𝑦 �𝑦𝑦23

Math 3 Unit 6 Worksheet 2

Math 3 Unit 6 Worksheet 2 Name: Multiplying & Dividing Radical Expressions Date: Per: [1-4] Simplify each expression.

1. √128𝑥𝑥53 2. √81𝑥𝑥73 3. �64𝑥𝑥6𝑦𝑦74 4. √32𝑥𝑥54

[5-19] Multiply and simplify, if possible, assuming all variable expressions are real numbers. 5. √43 ∙ √163 6. √5 ∙ √50 7. √93 ∙ √93 8. √3 ∙ √−4 9. √−123 ∙ √−183 10. √23 ∙ √7 11. √8𝑥𝑥∙�6𝑥𝑥𝑦𝑦2 12. 3�16𝑥𝑥4𝑦𝑦3 ∙ 2�𝑥𝑥𝑦𝑦23 13. 𝑥𝑥 �27𝑥𝑥2𝑦𝑦3 ∙ 2𝑥𝑥 √𝑥𝑥33 14. 5√2𝑐𝑐𝑑𝑑6 ∙ √2𝑐𝑐3𝑑𝑑 15. √𝑎𝑎5𝑏𝑏5 ∙ 3√2𝑎𝑎7𝑏𝑏6 16. √18𝑐𝑐94 ∙ √9𝑐𝑐𝑏𝑏44 17. −�2𝑥𝑥3𝑦𝑦23 ∙ 4�12𝑥𝑥5𝑦𝑦3 18. √2(√50 + 7) 19. √6(√6 + √18) 20. For the multiplication �2𝑥𝑥𝑦𝑦 ∙ �3𝑥𝑥3𝑦𝑦5, where x and y are real numbers, state the possible positive/negative configurations of the variables and explain the reasoning.

Math 3 Unit 6 Worksheet 2

[21-30] Simplify each expression. Rationalize all denominators. Assume all variables represent positive numbers.

21. √5√10

22. 1√53 23.

2√63 24.

1√84

25. 7√34√6𝑎𝑎

26. 1

√16𝑐𝑐3 27. 8√25𝑥𝑥23

28. √6𝑥𝑥4

�5𝑥𝑥2𝑦𝑦5 29.

2�7𝑥𝑥3𝑦𝑦−3�12𝑥𝑥4𝑦𝑦

30. √123

�6𝑥𝑥2𝑦𝑦3

31. Determine whether each expression is equivalent to √81𝑥𝑥73 . Select Yes or No for each expression.

Yes No 3�3𝑥𝑥73

9�𝑥𝑥73

3𝑥𝑥2 √3𝑥𝑥3

3𝑥𝑥 �3𝑥𝑥43

32. Determine whether each expression is equivalent to √20𝑥𝑥 ∙ �10𝑥𝑥3𝑦𝑦5. Select Yes or No for each expression.

Yes No 2√5𝑥𝑥 ∙ 𝑦𝑦�10𝑥𝑥3𝑦𝑦3

|𝑥𝑥2|�200𝑦𝑦5

2𝑦𝑦2�10𝑥𝑥4

10(𝑥𝑥𝑦𝑦)2�2𝑦𝑦

Math 3 Unit 6 Worksheet 3

Math 3 Unit 6 Worksheet 3 Name: Adding & Subtracting Radical Expressions Date: Per: Simplify, if possible. Assume all variables represent positive real numbers.

1. 10√5 + 2√5 2. 9√2 + 5√23 3. 5√11𝑥𝑥 − 8√11𝑥𝑥

4. 4√5 + 5√7 5. 9√𝑥𝑥23 − 4√𝑥𝑥23 6. 3√543 − 8√543

7. 5√3 + √12 8. 11√50 + √8 9. √163 + √543

10. 5√813 − 3√543 11. √72 + √18 + √50 12. √75 + 6√27 − 2√3

13. 4√18 + 6√75 − 2√48 14. √163 + 5√1283 − 2√543 15. 7√8𝑥𝑥 − 2√98𝑥𝑥

16. 4√9𝑥𝑥 − 2√𝑥𝑥 17. 4√216𝑤𝑤2 + 3√54𝑤𝑤2 18. √12𝑥𝑥33 + 4√27𝑥𝑥3

Math 3 Unit 6 Worksheet 3

19. √25𝑥𝑥5 + 3𝑥𝑥2√49𝑥𝑥 20. 𝑥𝑥√40𝑥𝑥43 − √135𝑥𝑥73 21. 𝑥𝑥√343𝑥𝑥3 + √729𝑥𝑥43

22. �4 − √6��4 + √6� 23. �3√7 + 5��3√7 − 5� 24. �√3 + √10��√3 − √10� 25. Determine whether each expression is equivalent to √50 + √8. Select Yes or No for each expression.

Yes No √58

5√2 + 2√2 9

(4 + 3)√2

26. Determine whether each expression is equivalent to 5√16𝑥𝑥 − 3√𝑥𝑥. Select Yes or No for each expression.

Yes No 20√𝑥𝑥 − 3√𝑥𝑥

17 2√15𝑥𝑥

17√𝑥𝑥

Math 3 Unit 6 Worksheet 4

Math 3 Unit 6 Worksheet 4 Name: Multiplying & Dividing Binomial Radical Expressions Date: Per: Simplify, if possible. Assume all variables represent positive real numbers.

1. �3 − 6√2��5 − 4√2� 2. �3 + √5��3 − √5� 3. �2 − √5�2

4. �2 + 5√7��5 + 4√7� 5. �5 + 4√3��1 − 2√3� 6. �9√5 + 7��9√5 − 7�

7. �√3 + √7�2 8. �2 − 4√3��2 + 4√3� 9. �1 + √98��5 + √2�

10. �3 + 2√3�2 11. �√7 − √3��√7 + √3� 12. �√𝑥𝑥 − √5��√𝑥𝑥 − 6√5�

13. �√12 + √50�2 14. �√1.25− √1.8��√5 − √0.2� 15. �√𝑥𝑥 + 2 − √𝑥𝑥 − 2��√𝑥𝑥 + 2 + √𝑥𝑥 − 2�

Math 3 Unit 6 Worksheet 4

16. 5

1+√2 17.

9√5−3

18. √3

8+√7

19. 6

4−√10 20.

12√23−9

21. √49

√9+√16

22. 2+√83√8−2

23. 2+3√279−√27

24. Find the perimeter and area of the rectangle 25. Find the unknown side and verify the area

4√5 − 7

2√5 + 4

???

√13 + 3

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 16

Math 3 Unit 6 Worksheet 5

Math 3 Unit 6 Worksheet 5 Name: Rational Exponents Date: Per: [1-14] Simplify each expression. 1. 81

12 2. 125

23 3. 16−

14 4. 9−

32 5. 3

12 ∙ 3

12

6. (−8)13 ∙ (−8)

13 7. (−32)0.8 8. 2431.25 9. 100−1.5 10. 16

12

1614

11. 100023

10012

12. 6413

813

13. �27−23�−1

14. (2532)

13

[15-19] Write each expression in radical form. 15. 𝑥𝑥

12 16. 𝑥𝑥

23 17. 𝑦𝑦1.25 18. 𝑦𝑦−

23 19. 𝑦𝑦−

34

[20- 24] Write each expression in exponential form. 20. √6 21. −�𝑦𝑦23 22. ��𝑦𝑦4 �

3 23. ��2𝑥𝑥𝑦𝑦3 �

6 24. √7𝑥𝑥

Math 3 Unit 6 Worksheet 5

[25-39] Write each expression in simplest form. Assume all variables represent positive real numbers.

25. (27𝑥𝑥9)13 26. (81𝑥𝑥12)−

14 27. �125𝑥𝑥6 𝑦𝑦

12�

23

28. �32𝑥𝑥10 𝑦𝑦12�

25 29. �8𝑥𝑥

34𝑦𝑦6�

43 30. 𝑥𝑥

23 ∙ 𝑥𝑥

13

31. 𝑦𝑦

12 ∙ 2𝑦𝑦

14 32. 4𝑥𝑥

25 ∙ 3𝑥𝑥

310 33. 𝑥𝑥

34 ÷ 𝑥𝑥

18

34. 𝑥𝑥12 𝑦𝑦−

12

𝑥𝑥14 𝑦𝑦−

32 35. � 𝑥𝑥

12

𝑦𝑦−34�8

36. �27𝑥𝑥9

8𝑦𝑦12�13

37. �18𝑥𝑥12

2𝑦𝑦4�12 38. � 2𝑥𝑥6

128𝑦𝑦15�23 39. �27𝑥𝑥

4

48𝑦𝑦2�32

Math 3 Unit 6 Worksheet 6

Math 3 Unit 6 Worksheet 6 Name: Solving Radical Equations Date: Per: [1-8] Solve the following for x. 1. 4√𝑥𝑥 + 3 = 15 2. √2𝑥𝑥 + 3 − 5 = 0 3. 5𝑥𝑥

23 = 45

4. 𝑓𝑓�𝑔𝑔(𝑥𝑥)� = 16 5. 𝑓𝑓�𝑔𝑔(𝑥𝑥)� = 27 6. 𝑓𝑓�𝑔𝑔(𝑥𝑥)� = 2

If 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥34 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 2 If 𝑓𝑓(𝑥𝑥) = 𝑥𝑥

32 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 1 If 𝑓𝑓(𝑥𝑥) = 𝑥𝑥

13 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 + 1

7. (𝑥𝑥 + 4)34 − 6 = 21 8. 2𝑥𝑥

12 + 4 = 6

[9-20] Solve the following for x. Check for extraneous solutions. 9. √𝑥𝑥 + 3 = 𝑥𝑥 + 1 10. √2𝑥𝑥 = √𝑥𝑥 + 5 11. 𝑥𝑥 − 6 = √3𝑥𝑥

Math 3 Unit 6 Worksheet 6

12. 𝑓𝑓(𝑥𝑥) − 𝑔𝑔(𝑥𝑥) = 0 13. √5 − 4𝑥𝑥 = −3 14. √4 − 11𝑥𝑥 = −𝑥𝑥 + 2 If 𝑓𝑓(𝑥𝑥) = √5 − 4𝑥𝑥 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥

15. 𝑓𝑓(𝑥𝑥) − 𝑔𝑔(𝑥𝑥) = 0 16. √𝑥𝑥 + 2 + 4 = 𝑥𝑥 17. √𝑥𝑥2 + 3 = 𝑥𝑥 + 1 If 𝑓𝑓(𝑥𝑥) = √1 − 𝑥𝑥 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 + 1

18. √𝑥𝑥 − √𝑥𝑥 − 5 = 2 19. √𝑥𝑥 + 9 = 1 + √2 + 𝑥𝑥 20. √𝑥𝑥 = √𝑥𝑥 − 8 + 2

Math 3 Unit 6 Worksheet 7

Math 3 Unit 6 Worksheet 7 Name: Graphing Radical Equations Date: Per: [1-6] Graph each function and state the domain and range.

1. 𝑦𝑦 = √𝑥𝑥 + 5 2. 𝑦𝑦 = √𝑥𝑥 + 3 3. 𝑦𝑦 = −√𝑥𝑥 − 2 Domain Domain Domain

Range Range Range 4. 𝑦𝑦 = 3√𝑥𝑥 + 1 − 6 5. 𝑦𝑦 = −2√𝑥𝑥 − 2 + 1 6. 𝑦𝑦 = 1

2 √𝑥𝑥 + 2 + 3 Domain Domain Domain

Range Range Range [7-11]: Sketch the graph for each of the following and find/solve for the indicated information.

7. 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 + 4 a) Domain & Range

b) 𝑥𝑥 − 𝑎𝑎𝑎𝑎𝑎𝑎 𝑦𝑦 − 𝑖𝑖𝑎𝑎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑠𝑠)

c) The open interval where 𝑓𝑓 is increasing d) The average rate of change for 𝑓𝑓 on 0 ≤ 𝑥𝑥 ≤ 9

Math 3 Unit 6 Worksheet 7

8. 𝑔𝑔(𝑥𝑥) = 5 − √𝑥𝑥 a) Domain & Range b) 𝑥𝑥 − 𝑎𝑎𝑎𝑎𝑎𝑎 𝑦𝑦 − 𝑖𝑖𝑎𝑎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑠𝑠) c) The open interval where 𝑔𝑔 is decreasing d) The open interval where 𝑔𝑔 is negative 9. ℎ(𝑥𝑥) = √𝑥𝑥 + 2 − 3 a) Domain & Range b) 𝑥𝑥 − 𝑎𝑎𝑎𝑎𝑎𝑎 𝑦𝑦 − 𝑖𝑖𝑎𝑎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑠𝑠) c) The open interval where ℎ is positive d) The average rate of change for ℎ on −1 ≤ 𝑥𝑥 ≤ 7 10. 𝑦𝑦 = 1

2 √𝑥𝑥 + 1 − 2 a) Domain & Range b) 𝑥𝑥 − 𝑎𝑎𝑎𝑎𝑎𝑎 𝑦𝑦 − 𝑖𝑖𝑎𝑎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑠𝑠) c) The open interval where the function is increasing d) The average rate of change for this function from 𝑥𝑥 = 3 to 𝑥𝑥 = 15 11. 𝑓𝑓(𝑥𝑥) = 3 − 2√𝑥𝑥 + 4 a) Domain & Range b) 𝑥𝑥 − 𝑎𝑎𝑎𝑎𝑎𝑎 𝑦𝑦 − 𝑖𝑖𝑎𝑎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑠𝑠) c) The average rate of change for 𝑓𝑓 from 𝑥𝑥 = −3 to 𝑥𝑥 = 5 d) The open interval where 𝑓𝑓 is positive

Math 3 Unit 6 Worksheet 8

Math 3 Unit 6 Worksheet 8 Name: Graphing Radical Equations Date: Per: [1-6] Graph each function and state the domain and range.

1. 𝑦𝑦 = √𝑥𝑥3 + 2 2. 𝑦𝑦 = √𝑥𝑥 − 33 3. 𝑦𝑦 = −√𝑥𝑥3 − 1 Domain Domain Domain

Range Range Range 4. 𝑦𝑦 = 2√𝑥𝑥 + 13 − 4 5. 𝑦𝑦 = −2√𝑥𝑥 − 23 + 1 6. 𝑦𝑦 = 1

2 √𝑥𝑥 + 23 + 3 Domain Domain Domain

Range Range Range

Math 3 Unit 6 Worksheet 8

[7-8]: Sketch the graph for each of the following and find/solve for the indicated information.

7. 𝑔𝑔(𝑥𝑥) = 1 − √𝑥𝑥 + 23 a) Domain & Range

b) 𝑥𝑥 − 𝑎𝑎𝑎𝑎𝑎𝑎 𝑦𝑦 − 𝑖𝑖𝑎𝑎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑠𝑠)

c) The open interval where 𝑔𝑔 is decreasing

d) The open interval where 𝑔𝑔 is negative

8. ℎ(𝑥𝑥) = 2√𝑥𝑥 − 33 + 2 a) Domain & Range

b) 𝑥𝑥 − 𝑎𝑎𝑎𝑎𝑎𝑎 𝑦𝑦 − 𝑖𝑖𝑎𝑎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑠𝑠)

c) The open interval where ℎ is negative

d) The average rate of change for ℎ on −5 ≤ 𝑥𝑥 ≤ 2

Math 3 Unit 6 Worksheet 9

Math 3 Unit 6 Worksheet 9 Name: Solving and Graphing Radical Equations Date: Per:

1. a) Solve algebraically for 𝑥𝑥: √𝑥𝑥 + 1 = 𝑥𝑥 − 1 b) Accurately graph the system of equations which is from the algebraic equation in part a) .

𝑓𝑓(𝑥𝑥) = √𝑥𝑥 + 1 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 1

𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) 𝑎𝑎𝑎𝑎 𝑥𝑥 = ____

2. a) Solve algebraically for 𝑥𝑥: −√𝑥𝑥 + 7 = −𝑥𝑥 − 1 b) Rewrite the algebraic equation into a system of

equations and accurately graph.

𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) 𝑎𝑎𝑎𝑎 𝑥𝑥 = ____

3. a) Solve algebraically for 𝑥𝑥: √𝑥𝑥 + 3 = 2𝑥𝑥 b) Rewrite the algebraic equation into a system of

equations and accurately graph.

𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) 𝑎𝑎𝑎𝑎 𝑥𝑥 = ____

4. a) Solve algebraically for 𝑥𝑥: −√𝑥𝑥 − 4 = 𝑥𝑥 − 10 b) Rewrite the algebraic equation into a system of

equations and accurately graph.

𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) 𝑎𝑎𝑎𝑎 𝑥𝑥 = ____

Math 3 Unit 6 Worksheet 9

[5-19] Solve the following for x. Check for extraneous solutions. 5. 1

3√𝑥𝑥 + 5 − 2 = 2 6. 17− 3√𝑥𝑥 − 4 = 2 7. 𝑓𝑓�𝑔𝑔(𝑥𝑥)� = 15 If 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 + 2 8. 4√𝑥𝑥 + 33 + 17 = 9 9. 2𝑓𝑓(𝑥𝑥) + 19 = 3 10. √8𝑥𝑥 + 9 = 3√𝑥𝑥 If 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 + 1 11. √10𝑥𝑥 + 8 = 2√3𝑥𝑥 12. 3√5𝑥𝑥 + 3 = 6√2𝑥𝑥 13. 2√𝑥𝑥 = √3𝑥𝑥 + 6 14. 𝑓𝑓(𝑔𝑔(𝑥𝑥)) = 𝑔𝑔(𝑥𝑥) − 2 15. √𝑥𝑥 − 2 = 4 − 𝑥𝑥 16. √2𝑥𝑥 + 5 = 𝑥𝑥 + 1 If 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 + 2 and 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 17. 3𝑥𝑥 − 5√𝑥𝑥 = 2 18. √2𝑥𝑥 + 5 = 2√2𝑥𝑥 + 1 19. √𝑥𝑥 + 7 = 𝑥𝑥 − 5

Math 3 Unit 6 Review Worksheet 1

Math 3 Unit 6 Review Worksheet 1 Name: Radical Functions Date: Per: [1-18] Simplify the following:

1. √49𝑚𝑚2 2. √27𝑚𝑚4 3. √121𝑚𝑚6𝑛𝑛8 4. √−273 5. √8𝑚𝑚33 6. √−325 7. √80𝑚𝑚4𝑛𝑛5 8. √147𝑚𝑚3𝑛𝑛4

9. 18√13+2

10. 1√43 11. 5

�25𝑥𝑥23 12. �20𝑥𝑥4𝑦𝑦2

�5𝑥𝑥𝑦𝑦3

13. 3𝑥𝑥√7𝑥𝑥

14. √8𝑥𝑥3 + 3𝑥𝑥√2𝑥𝑥 15. 4𝑚𝑚√75𝑚𝑚𝑛𝑛6 − 2𝑛𝑛2√48𝑚𝑚3𝑛𝑛2

16. 7�2𝑥𝑥3𝑦𝑦6 ∙ �2𝑥𝑥𝑦𝑦 17. �27𝑥𝑥2𝑦𝑦 ∙ �𝑥𝑥3𝑦𝑦5 18. −3

√7 − 5

Math 3 Unit 6 Review Worksheet 1

19. Rewrite the functions below so they are translated 5 units left and 7 units up. Sketch the translated function. a) 𝑦𝑦 = 𝑥𝑥2 b) 𝑦𝑦 = (𝑥𝑥 − 2)2 + 3 c) 𝑦𝑦 = |𝑥𝑥| d) 𝑦𝑦 = |𝑥𝑥 + 2| + 1 e) 𝑦𝑦 = √𝑥𝑥 f) 𝑦𝑦 = √𝑥𝑥 + 1 − 2 g) 𝑦𝑦 = √𝑥𝑥3 h) 𝑦𝑦 = √𝑥𝑥 − 13 − 1

20. Given: 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 − 2)2 + 5 , 𝑥𝑥 ≥ 2 . Sketch 𝑓𝑓(𝑥𝑥) and identify its domain and range.

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

Math 3 Unit 6 Review Worksheet 1

21. Given: 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 3

Sketch 𝑦𝑦 = −𝑓𝑓(𝑔𝑔(𝑥𝑥)) and identify its domain and range.

[22-24] Solve the following equation for x: 22. 1

4 √𝑥𝑥 + 3 − 1 = 0 23. √2𝑥𝑥 + 10 = √4𝑥𝑥 24. √2𝑥𝑥2 + 16 = 2√3𝑥𝑥 25. a) Solve the following equation algebraically for 𝑥𝑥: √𝑥𝑥 − 2 = 𝑥𝑥 − 4 b) Rewrite the above equation as a system of two equations and graph. 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) = For 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥), 𝑥𝑥 =

Math 3 Unit 6 Review Worksheet 1

[26-28] Graph the following and find the following: a) x-intercept and y-intercept b) Domain and Range c) the open interval where f is increasing d) the open interval where f is decreasing e) the open interval where f is negative f) the open interval where f is positive g) Average rate of change on the specified interval

• [3, 7] for problem #6 • [−2, 1] for problem #7 • [−4, 4] for problem #8

26. 𝑓𝑓(𝑥𝑥) = 12 √𝑥𝑥 − 3 − 1

27. 𝑓𝑓(𝑥𝑥) = 4 − √𝑥𝑥 + 3 28. 𝑓𝑓(𝑥𝑥) = 1

2 √𝑥𝑥 + 43 − 1

Math 3 Unit 6 Review Worksheet 1

29. Determine whether each expression is equivalent to �32𝑥𝑥3𝑦𝑦2. Select Yes or No for each expression.

30. Selected Response: Given 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 − 4 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 + 2, which of the following is true? Choose ALL that apply.

(A) 𝑓𝑓(𝑥𝑥)𝑔𝑔(𝑥𝑥)

= 𝑥𝑥 − 2, 𝑥𝑥 ≠ −2 (B) 𝑓𝑓�𝑔𝑔(𝑥𝑥)� = 𝑥𝑥2 + 4𝑥𝑥 (C) (𝑔𝑔 ∙ 𝑓𝑓)(𝑥𝑥) = 𝑥𝑥2 − 2

[31-35]: Matching. 31. 𝑦𝑦 = √𝑥𝑥 − 2 − 1 32. 𝑦𝑦 = −√𝑥𝑥 − 2 − 1 33. 𝑦𝑦 = √𝑥𝑥 − 23 − 1 34. 𝑦𝑦 = −√𝑥𝑥 + 2 − 1

35. 𝑦𝑦 = −√𝑥𝑥 + 23 − 1

Yes No 4𝑥𝑥�2𝑥𝑥𝑦𝑦2

22𝑥𝑥𝑦𝑦√2𝑥𝑥

4𝑥𝑥|𝑦𝑦|√2𝑥𝑥

2𝑥𝑥�2𝑥𝑥𝑦𝑦2

2𝑥𝑥|𝑦𝑦|√8𝑥𝑥

A)

C) D)

E)

H)

F)

B)

G)

Math 3 Unit 6 Review Worksheet 1

[36-38]: Solve for x. 36. 75 − 4√𝑥𝑥 − 5 = 39 37. 3√5𝑥𝑥 + 4 = 6√2𝑥𝑥 38. √𝑥𝑥 + 3 = −𝑥𝑥 − 1 39. Go back to problem #38. Rewrite the equation as a system of two equations and graph both on the coordinate axes to the right. Find the solution(s) to the system based on the graph. 40. Given 𝑓𝑓(𝑥𝑥) = 2√𝑥𝑥 + 5 − 4 a) Sketch. b) Identify domain & range. c) Find x-int & y-int. d) Identify open interval where f is increasing. e) Identify open interval where f is decreasing. f) Identify open interval where f is positive. g) Identify open interval where f is negative. h) Find the average rate of change for f on −4 ≤ 𝑥𝑥 ≤ 11. 41. If 𝑓𝑓(𝑥𝑥) = −√𝑥𝑥 − 2 − 7, find its domain & range.

Math 3 Unit 6 Review Worksheet 2

Math 3 Unit 6 Review Worksheet 2 Name: Radical Functions Date: Per: [1-6] Selected Response: Choose all answers that apply 1. Choose which of the following expression(s) is equivalent to �81𝑥𝑥3𝑦𝑦2 1. {Hint: There are two correct responses} A) 3𝑥𝑥|𝑦𝑦|√𝑥𝑥 B) 32𝑥𝑥�𝑥𝑥𝑦𝑦2 C) 9𝑦𝑦√𝑥𝑥3 D) 9𝑥𝑥|𝑦𝑦|√𝑥𝑥 E) 9𝑥𝑥𝑦𝑦√𝑥𝑥

2. Choose which of the following expression(s) is equivalent to √16 2. {Hint: There are three correct responses} A) – 8 B) 8 C) (–2)2 D) 22 E) 4 3. Choose which of the following expression(s) is equivalent to 4 12x x 3. {Hint: There are three correct responses} A) 8𝑥𝑥√3𝑥𝑥 B) 24√𝑥𝑥3 C) 6𝑥𝑥√3𝑥𝑥 + 2𝑥𝑥√3𝑥𝑥 D) 8√3𝑥𝑥3 E) 12𝑥𝑥√2𝑥𝑥

4. Choose which of the following expression(s) is equivalent to �−8𝑥𝑥3𝑦𝑦63 4. {Hint: There are two correct responses} A) 2|𝑥𝑥|𝑦𝑦2 B) −2|𝑥𝑥|𝑦𝑦2 C) 2𝑥𝑥𝑦𝑦3 D) −2𝑥𝑥𝑦𝑦2 E) −4

2𝑥𝑥𝑦𝑦3

5. Choose which of the following expression(s) is equivalent to √8𝑥𝑥 + √2𝑥𝑥 5. {Sorry, no hints this time.} A) 3√2𝑥𝑥 B) 2√2𝑥𝑥 + √2𝑥𝑥 C) √10𝑥𝑥 D) √10𝑥𝑥2 E) |𝑥𝑥|√10

6. Choose which of the following expression(s) is equivalent to (4 − √2)(2 + √2) 6. {Sorry, no hints this time.} A) 8 + 4√2 − 2√2 − 2 B) 6 C) 8 + √8 − √4 − √4 D) 2(3 + √2) E) 6 + 2√2

Math 3 Unit 6 Review Worksheet 2

7. Graph: 𝑓𝑓(𝑥𝑥) = −√𝑥𝑥 + 5 + 3 A. Sketch the graph

B. Domain: Range

C. Identify the x-intercept: show work

D. Identify the y-intercept:

E. Open interval where 𝑓𝑓 is increasing

F. Open interval where 𝑓𝑓 is positive

G. Identify the rate of change over −4 ≤ 𝑥𝑥 ≤ 44 show work

H. Rewrite 𝑓𝑓 if it were shifted 7 units to the left and 2 units down 𝑓𝑓(𝑥𝑥) =

Original Vertex:

Shift:

New Vertex:

8. Rationalize: 207√3

9. Rationalize: 3

7−√2 8.

9.

10. Simplify: 2√10𝑎𝑎𝑎𝑎 ∙ 6√14𝑎𝑎𝑎𝑎2 11. Rationalize: 8 √7 3

2 √253 10.

11.

Math 3 Unit 6 Review Worksheet 2

12. Simplify: 5𝑎𝑎√80𝑎𝑎33 + 3𝑎𝑎2√103 12.

13. Simplify: 3𝑎𝑎5 �8𝑥𝑥7

𝑎𝑎�6𝑥𝑥8 13.

14. Cube root graphing: A. Graph 𝑓𝑓(𝑥𝑥) = − √𝑥𝑥 + 33 − 5

B. Domain Range

C. This function is: always increasing / always decreasing / at times increasing and at times decreasing

(Circle the correct answer)

D. Identify the x-intercept: show work

E. Identify the y-intercept:

F. Open interval where 𝑓𝑓 is positive G. Open interval where 𝑓𝑓 is negative

15. Multiply: (2 + 3√5)(5 − 2√5) 16. Simplify: �8 − 3√2�2

15. 16.

Math 3 Unit 6 Review Worksheet 2

Math 3 Unit 6 Selected Answers

Selected Answers to Math 3 Unit 6 Worksheet #7:

7 𝑎𝑎) 𝐷𝐷: 𝑥𝑥 ≥ 0 𝑜𝑜𝑜𝑜 [0,∞);𝑅𝑅: 𝑦𝑦 ≥ 4 𝑜𝑜𝑜𝑜 [4,∞) 𝑏𝑏) (0, 4) 𝑐𝑐) 𝑥𝑥 ≥ 0 𝑜𝑜𝑜𝑜 (0,∞) 𝑑𝑑) 1/3

9 𝑎𝑎) 𝐷𝐷: 𝑥𝑥 ≥ −2 𝑜𝑜𝑜𝑜 [−2,∞);𝑅𝑅:𝑦𝑦 ≥ −3 𝑜𝑜𝑜𝑜 [−3,∞) 𝑏𝑏) (7, 0) 𝑎𝑎𝑎𝑎𝑑𝑑 (0,√2 − 3) ≈ (0,−1.586) 𝑐𝑐) 𝑥𝑥 > 7 𝑜𝑜𝑜𝑜 (7,∞) 𝑑𝑑) 1/4

11 𝑎𝑎) 𝐷𝐷:𝑥𝑥 ≥ −4 𝑜𝑜𝑜𝑜 [−4,∞);𝑅𝑅: 𝑦𝑦 ≤ −3 𝑜𝑜𝑜𝑜 (−∞,−3] 𝑏𝑏) (−7/4, 0) 𝑎𝑎𝑎𝑎𝑑𝑑 (0,−1) 𝑐𝑐) − 1/2 𝑑𝑑) − 4 < 𝑥𝑥 < −7/4 𝑜𝑜𝑜𝑜 (−4,−7/4)

Selected Answers to Math 3 Unit 6 Worksheet #1: 1. ±5 3. ± 7

11 5. – 2 7. 6 9. 0.1 11. |𝑎𝑎3|𝑏𝑏6 13. 4|𝑥𝑥|𝑦𝑦4 15. −27𝑥𝑥4|𝑦𝑦|√𝑥𝑥 17. 3𝑎𝑎3|𝑏𝑏3|√3𝑎𝑎

19. 5𝑎𝑎4𝑏𝑏3√3𝑏𝑏 21. 3𝑥𝑥4𝑦𝑦5 23. 𝑥𝑥4𝑦𝑦 �𝑥𝑥2𝑦𝑦23 25. −4𝑥𝑥2𝑦𝑦 �3𝑦𝑦23 27. 2𝑥𝑥𝑦𝑦3 √2𝑥𝑥3 29. −3𝑎𝑎2𝑏𝑏3𝑐𝑐2 √−2𝑏𝑏2𝑐𝑐3 31. 𝑥𝑥 = 2 33. 𝑥𝑥 = −3 Selected Answers to Math 3 Unit 6 Worksheet #2: 1. 4𝑥𝑥√2𝑥𝑥23 3. 2|𝑥𝑥|𝑦𝑦 �4𝑥𝑥2𝑦𝑦34 5. 4 7. 3√33 9. 6 11. 4𝑥𝑥|𝑦𝑦|√3 13. 6𝑥𝑥3 �3𝑥𝑥2𝑦𝑦3 15. 𝑥𝑥2𝑦𝑦3√6

17. 3𝑐𝑐2|𝑏𝑏|√2𝑐𝑐24 19. 10 + 7√2 21. √22

23. √363

3 25. 7√2𝑎𝑎

8𝑎𝑎 27. 8 √5𝑥𝑥

3

5𝑥𝑥 29. −√21𝑥𝑥

9𝑥𝑥

Selected Answers to Math 3 Unit 6 Worksheet #3: 1. 12√5 3. −3√11𝑥𝑥 5. 5√𝑥𝑥23 7. 7√3 9. 5√23 11. 14√2 13. 12√2 + 22√3 15. 0 17. 33𝑤𝑤√6 19. 26𝑥𝑥2√𝑥𝑥 21. 16𝑥𝑥 √𝑥𝑥3 23. 38 Selected Answers to Math 3 Unit 6 Worksheet #4: 1. 63− 42√2 3. 9 − 4√5 5. −19 − 6√3 7. 10 + 2√21 9. 19 + 36√2 11. 4 13. 62 + 20√6 15. 4

17. 9√5+27−4

19. 4 + √10 21. 1 23. 33+29√318

25. ? ? ? = 4√13 − 12 Selected Answers to Math 3 Unit 6 Worksheet #5: 1. 9 3. ½ 5. 3 7. 16 9. 1

1000 11. 10 13. 9 15. √𝑥𝑥 17. �𝑦𝑦54 19. 1

�𝑦𝑦34 21. −𝑦𝑦23 23. (2𝑥𝑥𝑦𝑦)2 25. 3𝑥𝑥3 27.

25𝑥𝑥4𝑦𝑦13 29. 16𝑥𝑥𝑦𝑦8 31. 2𝑦𝑦

34 33. 𝑥𝑥

58 35. 𝑥𝑥4𝑦𝑦6 37. 3𝑥𝑥

6

𝑦𝑦2 39. 27𝑥𝑥

6

64𝑦𝑦3

Selected Answers to Math 3 Unit 6 Worksheet #6: 1. 𝑥𝑥 = 9 3. 𝑥𝑥 = ±27 5. 𝑥𝑥 = 10 7. 𝑥𝑥 = 77 9. 𝑥𝑥 = 1 11. 𝑥𝑥 = 12 13. ∅ 15. 𝑥𝑥 = 0 17. 𝑥𝑥 = 1 19. 𝑥𝑥 = 7

Selected Answers to Math 3 Unit 6 Worksheet #9: 1. {3}; 0 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑒𝑒𝑥𝑥𝑒𝑒𝑜𝑜𝑎𝑎𝑎𝑎𝑒𝑒𝑜𝑜𝑒𝑒𝑖𝑖 𝑜𝑜𝑜𝑜𝑜𝑜𝑒𝑒 3. {1};−3

4 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑒𝑒𝑥𝑥𝑒𝑒𝑜𝑜𝑎𝑎𝑎𝑎𝑒𝑒𝑜𝑜𝑒𝑒𝑖𝑖 𝑜𝑜𝑜𝑜𝑜𝑜𝑒𝑒 5. {139} 7. {223} 9. ∅ 11. {4} 13. {6} 15.

{3 𝑜𝑜𝑎𝑎𝑜𝑜𝑦𝑦} 17. {4 𝑜𝑜𝑎𝑎𝑜𝑜𝑦𝑦} 19. {9 𝑜𝑜𝑎𝑎𝑜𝑜𝑦𝑦} Selected Answers to Math 3 Unit 6 Worksheet #10: 1. 𝑥𝑥2 + 4𝑥𝑥 − 12; ℝ 3. −𝑥𝑥2 − 2𝑥𝑥 + 8; ℝ 5. 𝑥𝑥 + 3√𝑥𝑥 − 10; [0,∞) 7. 1

5 9. 1 11. −𝑥𝑥2 + 2𝑥𝑥 − 4

13. −(2𝑥𝑥 − 4)2 15. (2𝑥𝑥 + 2)2 17. (𝑎𝑎 + 1)2

Selected Answers to Math 3 Unit 6 Worksheet #8:

7 𝑎𝑎) 𝐷𝐷 & 𝑅𝑅: 𝑎𝑎𝑜𝑜𝑜𝑜 𝑜𝑜𝑒𝑒𝑎𝑎𝑜𝑜𝑖𝑖 𝑜𝑜𝑜𝑜 (−∞,∞) 𝑏𝑏) (−1,0); (0, 1 − √23 ) ≈ (0,−0.260) 𝑐𝑐) 𝑎𝑎𝑜𝑜𝑜𝑜 𝑜𝑜𝑒𝑒𝑎𝑎𝑜𝑜𝑖𝑖 𝑜𝑜𝑜𝑜 (−∞,∞) 𝑑𝑑) 𝑥𝑥 > −1 𝑜𝑜𝑜𝑜 (−1,∞)

8 𝑎𝑎) 𝐷𝐷 & 𝑅𝑅: 𝑎𝑎𝑜𝑜𝑜𝑜 𝑜𝑜𝑒𝑒𝑎𝑎𝑜𝑜𝑖𝑖 𝑜𝑜𝑜𝑜 (−∞,∞) 𝑏𝑏) (2, 0) 𝑎𝑎𝑎𝑎𝑑𝑑 (0, 2 − 2√33 ) ≈ (0,−0.884) 𝑐𝑐) 𝑥𝑥 < 2 𝑜𝑜𝑜𝑜 (−∞, 2) 𝑑𝑑) 27

Math 3 Unit 6 Selected Answers

Selected Answers to Math 3 Unit 6 Worksheet #11:

7. a) 𝑦𝑦 = ±�12

(𝑥𝑥 − 2) b) 𝐷𝐷: ℝ; 𝑅𝑅: [2,∞) c) 𝐷𝐷: [2,∞); 𝑅𝑅: ℝ d) Inverse is not a function

9. a) 𝑦𝑦 = ±√−𝑥𝑥 + 2 b) 𝐷𝐷: (−∞,∞); 𝑅𝑅: (−∞, 0] c) 𝐷𝐷: (−∞, 0]; 𝑅𝑅: (−∞,∞) d) Inverse is not a function 11. a) 𝑦𝑦 = 1𝑥𝑥

b) 𝐷𝐷: 𝑥𝑥 ≠ 0; 𝑅𝑅:𝑦𝑦 ≠ 0 c) 𝐷𝐷: 𝑥𝑥 ≠ 0; 𝑅𝑅:𝑦𝑦 ≠ 0 d) Inverse is a function 13. a) 𝑦𝑦 = − 1𝑥𝑥

+ 3 b) 52 c) 1 d) x

Selected Answers to Math 3 Unit 6 Review WS 2: 1. BD 2. CDE 3. ACD 4. DE 5. AB 6. ADE 7. {86} 8. {4/3} 9. {−2}; 1 is extraneous 11. 𝑥𝑥2 − 𝑥𝑥 − 6 12. 1

2𝑥𝑥+1 , 𝑥𝑥 ≠ 3 13. 2𝑥𝑥2 + 3𝑥𝑥 − 5

14. b. 𝐷𝐷 = [−5,∞) & 𝑅𝑅 = [−4,∞) c. 𝑥𝑥 − 𝑖𝑖𝑎𝑎𝑒𝑒 = (−1,0) & 𝑦𝑦 − 𝑖𝑖𝑎𝑎𝑒𝑒 = (0,2√5− 4) d. (−5,∞) e. ∅ f. (−1,∞) g. (−5,−1)

h. 𝐴𝐴𝐴𝐴𝐴𝐴 𝑜𝑜𝑎𝑎𝑒𝑒𝑒𝑒 𝑜𝑜𝑜𝑜 𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝐴𝐴𝑒𝑒 𝑜𝑜𝑎𝑎 [−4, 11] = 2/5

15. a. 𝑜𝑜(𝑥𝑥): 𝐷𝐷 = [2,∞) & 𝑅𝑅 = (−∞,−7]

b. 𝑜𝑜−1(𝑥𝑥): 𝐷𝐷 = (−∞,−7] & 𝑅𝑅 = [2,∞) c. 𝑜𝑜−1(𝑥𝑥) = (𝑥𝑥 + 7)2 + 2; 𝑥𝑥 ≤ −7

Selected Answers to Math 3 Unit 6 Review WS 1: 25. {86} 26. {4/3} 27. {−2}; 1 is extraneous 29. 𝑥𝑥2 − 𝑥𝑥 − 6 30. 1

2𝑥𝑥+1 , 𝑥𝑥 ≠ 3 31. 2𝑥𝑥2 + 3𝑥𝑥 − 5 32. b. 𝐷𝐷 =

[−5,∞) & 𝑅𝑅 = [−4,∞) c. 𝑥𝑥 − 𝑖𝑖𝑎𝑎𝑒𝑒 = (−1,0) & 𝑦𝑦 − 𝑖𝑖𝑎𝑎𝑒𝑒 = (0,2√5 − 4) d. (−5,∞) e. ∅ f. (−1,∞) g. (−5,−1) h. 𝐴𝐴𝐴𝐴𝐴𝐴 𝑜𝑜𝑎𝑎𝑒𝑒𝑒𝑒 𝑜𝑜𝑜𝑜 𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝐴𝐴𝑒𝑒 𝑜𝑜𝑎𝑎 [−4, 11] = 2/5 33. a. 𝑜𝑜(𝑥𝑥): 𝐷𝐷 = [2,∞) & 𝑅𝑅 = (−∞,−7] b. 𝑜𝑜−1(𝑥𝑥): 𝐷𝐷 = (−∞,−7] & 𝑅𝑅 = [2,∞) c. 𝑜𝑜−1(𝑥𝑥) = (𝑥𝑥 + 7)2 + 2; 𝑥𝑥 ≤ −7