Find the first 3 iterates of the function [tex]f(x)=-2 x+3[/tex] when [tex]x_0=0[/tex].
A. 1,1,1
B. 2,-1, 5
C. -2,0,2
D. 3,-3,9
Answer (2)
We are given the function f ( x ) = − 2 x + 3 and the initial value x 0 = 0 .
Calculate the first iterate: x 1 = f ( x 0 ) = − 2 ( 0 ) + 3 = 3 .
Calculate the second iterate: x 2 = f ( x 1 ) = − 2 ( 3 ) + 3 = − 3 .
Calculate the third iterate: x 3 = f ( x 2 ) = − 2 ( − 3 ) + 3 = 9 . The first three iterates are 3 , − 3 , 9 .
Explanation
Understanding the Problem We are given the function f ( x ) = − 2 x + 3 and the initial value x 0 = 0 . We need to find the first three iterates of the function, which are x 1 = f ( x 0 ) , x 2 = f ( x 1 ) , and x 3 = f ( x 2 ) .
Calculating the First Iterate First, we calculate x 1 = f ( x 0 ) = f ( 0 ) . Substituting x 0 = 0 into the function, we get: x 1 = − 2 ( 0 ) + 3 = 3
Calculating the Second Iterate Next, we calculate x 2 = f ( x 1 ) = f ( 3 ) . Substituting x 1 = 3 into the function, we get: x 2 = − 2 ( 3 ) + 3 = − 6 + 3 = − 3
Calculating the Third Iterate Finally, we calculate x 3 = f ( x 2 ) = f ( − 3 ) . Substituting x 2 = − 3 into the function, we get: x 3 = − 2 ( − 3 ) + 3 = 6 + 3 = 9
Listing the Iterates Therefore, the first three iterates of the function are 3 , − 3 , 9 .
Examples
Understanding function iteration is crucial in many fields. For example, in computer graphics, iterative functions are used to generate fractals, complex and beautiful images created by repeatedly applying a simple mathematical formula. Similarly, in economics, iterative models can simulate how markets evolve over time based on initial conditions and response functions. By understanding how functions iterate, you can model and predict the behavior of complex systems in various real-world scenarios.
The first three iterates of the function f ( x ) = − 2 x + 3 starting from x 0 = 0 are 3, -3, and 9. Therefore, the correct answer is D. 3, -3, 9.
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