# c and Those complex num black, while those that lead to un used similar ideas to generate his own recurs The image "i of the storm" reproduced here function the is based f(2) = (1-i)z*+(7+1)z where again z is a 2z +6 %3D variable that will be replaced with complex numbers. The image is actually a picture of the complex plane, with the origin in the very center of the golden ring. The golden ring consists of those complex numbers that lie a distance between 0.9 and 1.1 units from the origin. The rules for coloring other complex numbers in the plane are as follows: given an initial complex number z not on the gold ring, f(z) is calculated. If the complex number f(z) lies somewhere on the gold ring, the original number z is colored the deepest shade of green. If not, the iterate f (z) is calculated. If this result lies in the gold ring, the original z is colored a bluish shade of green. If not, the process continues up to the 12th iterate f (z), using a different color time. If f" (z) lies in the gold ring, z is colored red, and if not the process halts and z is colored black. The idea of recursion can be used to generate any number of similar images, with the end result usually striking and often surprising even to the creator. Exercises In each of the following problems, use the information given to determine a. (f + g)(-1). f b. (f -g)(-1),c. (fg)(-1), and d. (4) -(-1). See Examples 1,2, and 3. 1. f(-1)=-3 and g(-1)=5 2. f(-1)=0 and g(-1)=-1 3. f(x)=x² – 3 and g(x)= x %3D 4. f(x)= Vx and g(x)= x – 1 5. f(-1)= 15 and g(-1)=-3 6. f(x)= and g(x)= 6x %3D %3D 7. f(x)=x* +1 and g(x) = x" +2 %3D 8. f(x)=, 6-X and g(x) = %3D 9. f={(5, 2).(0, – 1).(-1, 3).(-2, 4)} and g = {(-1, 3).(0, 5)} -4 %3D 10. f = {(3,15).(2, – 1).(-1, 1)} and g(x)= -2 283 Combining Functions Section 3.6 11. 12. 4. 4 2 -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 2 4 13. 14. 4 4 0+ 0+ -2 -2 -4 -4 -4 -2 0 -4 -2 4 In each of the following problems, find a. the formula and domain for f+ g, and b. the formula and domain for See Examples 2 and 3. 15. f(x)=|x| and g (x) = Vx 16. f(x) = x² -1 and g(x)= {x %3D 3 17. f(x)=x-1 and g(x) = x² – 1 18. f(x)= x² and g (x)= x – 3 19. f(x) = 3x and g(x)= x' –8 20. f(x) = x² +4 and g (x) = /x – 2 2 21. f(x)=-2x² and g(x)=|x+4| 22. f (x)= 6x – 1 and g (x) = x³ 4- ఉం 2. 2. 2.

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c and Those complex num black, while those that lead to un used similar ideas to generate his own recurs The image "i of the storm" reproduced here function the is based f(2) = (1-i)z*+(7+1)z where again z is a 2z +6 %3D variable that will be replaced with complex numbers. The image is actually a picture of the complex plane, with the origin in the very center of the golden ring. The golden ring consists of those complex numbers that lie a distance between 0.9 and 1.1 units from the origin. The rules for coloring other complex numbers in the plane are as follows: given an initial complex number z not on the gold ring, f(z) is calculated. If the complex number f(z) lies somewhere on the gold ring, the original number z is colored the deepest shade of green. If not, the iterate f (z) is calculated. If this result lies in the gold ring, the original z is colored a bluish shade of green. If not, the process continues up to the 12th iterate f (z), using a different color time. If f" (z) lies in the gold ring, z is colored red, and if not the process halts and z is colored black. The idea of recursion can be used to generate any number of similar images, with the end result usually striking and often surprising even to the creator. Exercises In each of the following problems, use the information given to determine a. (f + g)(-1). f b. (f -g)(-1),c. (fg)(-1), and d. (4) -(-1). See Examples 1,2, and 3. 1. f(-1)=-3 and g(-1)=5 2. f(-1)=0 and g(-1)=-1 3. f(x)=x² – 3 and g(x)= x %3D 4. f(x)= Vx and g(x)= x – 1 5. f(-1)= 15 and g(-1)=-3 6. f(x)= and g(x)= 6x %3D %3D 7. f(x)=x* +1 and g(x) = x" +2 %3D 8. f(x)=, 6-X and g(x) = %3D 9. f={(5, 2).(0, – 1).(-1, 3).(-2, 4)} and g = {(-1, 3).(0, 5)} -4 %3D 10. f = {(3,15).(2, – 1).(-1, 1)} and g(x)= -2

283 Combining Functions Section 3.6 11. 12. 4. 4 2 -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 2 4 13. 14. 4 4 0+ 0+ -2 -2 -4 -4 -4 -2 0 -4 -2 4 In each of the following problems, find a. the formula and domain for f+ g, and b. the formula and domain for See Examples 2 and 3. 15. f(x)=|x| and g (x) = Vx 16. f(x) = x² -1 and g(x)= {x %3D 3 17. f(x)=x-1 and g(x) = x² – 1 18. f(x)= x² and g (x)= x – 3 19. f(x) = 3x and g(x)= x' –8 20. f(x) = x² +4 and g (x) = /x – 2 2 21. f(x)=-2x² and g(x)=|x+4| 22. f (x)= 6x – 1 and g (x) = x³ 4- ఉం 2. 2. 2.