Using the function g(x) = −2 |2 - x|+ 4 and the initial value of 1.72, what is the loop that forms after many iterations?
My answers are 2.24, 3.52, 0.96, 1.92, 3.84, 0.32, 0.64, 1.28, 2.56, and 2.88 I'm not exactly sure if this is correct because I feel like there are too many loops so if this is wrong can someone please explain to me why these are wrong?
Hmmm. I don't see any theoretical solution, but after 55 iterations g(x)=4, so that the next time
g(x) = 4-2|2-4| = 4-2*2 = 0
g(0) = 4-2|2-0| = 4-2*2 = 0
and it stays zero
Maybe roundoff error contributes to this result.
To determine the loop that forms after many iterations using the given function g(x) = −2 |2 - x|+ 4 and the initial value of 1.72, we need to repeatedly apply the function to the outputs.
Let's begin by finding g(1.72):
g(1.72) = −2 |2 - 1.72| + 4
= −2 |0.28| + 4
= −2 (0.28) + 4
= -0.56 + 4
= 3.44
Now, let's find g(3.44):
g(3.44) = −2 |2 - 3.44| + 4
= −2 |-1.44| + 4
= −2 (1.44) + 4
= -2.88 + 4
= 1.12
Next, let's find g(1.12):
g(1.12) = −2 |2 - 1.12| + 4
= −2 |0.88| + 4
= −2 (0.88) + 4
= -1.76 + 4
= 2.24
Continuing this process, we can find subsequent iterations:
g(2.24) = −2 |2 - 2.24| + 4 = 2.24 (Loop formed)
g(2.24) = −2 |2 - 2.24| + 4 = 2.24 (Loop continues)
...
Based on the calculations, it appears that the loop indeed forms starting from 2.24. Therefore, the loop is 2.24, 2.24, 2.24,...
The values you provided (2.24, 3.52, 0.96, 1.92, 3.84, 0.32, 0.64, 1.28, 2.56, and 2.88) seem to be a mix of values from different iterations and not part of the loop. Consequently, this would explain why you feel there are too many loops.
Remember, to find the loop after many iterations, you need to repeatedly apply the function to the output of each iteration until a repeating pattern emerges.