Question

If f(z) = z2 + c is iterated with an initial value of 2 + 3i and z1 = -1 + 15i, find c. a. 7 - 4i c. 4 - 3i b. 7 + 4i d. 4 + 3i Please select the best answer from the choices provided A B C D

Answer 1

Answer: choice D) 4 + 3i

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Plug in the initial value z = 2 + 3i. Simplify.

f(z) = z^2 + c

f(2+3i) = (2+3i)^2 + c

f(2+3i) = (2+3i)*(2+3i) + c

f(2+3i) = 2(2+3i) + 3i(2+3i) + c

f(2+3i) = 4 + 6i + 6i + 9i^2 + c

f(2+3i) = 4 + 12i + 9i^2 + c

f(2+3i) = 4 + 12i + 9(-1) + c

f(2+3i) = 4 + 12i - 9 + c

f(2+3i) = -5 + 12i + c

The initial z value is z = 2+3i, which when plugged into f(z) leads to the term z1 = -1+15i. The input is 2+3i and the output is -1+15i

Set f(2+3i) equal to the output z1 = -1+15i and solve for c

f(2+3i) = z1

-5 + 12i + c = -1 + 15i

c = -1 + 15i + 5 - 12i

c = 4 + 3i

So if c = 4 + 3i, then f(z) = z^2 + c will have f(2+3i) = -1 + 15i