Come up with a new linear function that has a slope that falls in the range
−1< m <0 . Choose two different initial values. For this new linear function,
what happens to the function’s values after many iterations? Are the
function’s values getting close to a particular number in each case?
PLEASE PLEASE PLEASE HELP I NEED IT
What was the old linear function?
What type of iteration are you doing?
The old iteration function is f(4)=½*4+1=3 with an initial value of 4. Use the previous output as your new input. IDK the actual answer.
To come up with a linear function with a slope in the range −1 < m < 0, we can start with the general form of a linear function:
f(x) = mx + b
Where "m" represents the slope and "b" represents the y-intercept. Since we want a slope that falls in the specified range, we can choose any value for "b" and select "m" to meet the criteria.
Let's consider two different initial values:
1. f1(x) = -0.5x + 2
- In this case, the slope "m" is -0.5, falling within the range −1 < m < 0.
- The initial y-intercept is 2 (b = 2).
2. f2(x) = -0.75x - 3
- Here, the slope "m" is -0.75, which also falls within the specified range.
- The initial y-intercept is -3 (b = -3).
Now, let's analyze what happens to the function's values after many iterations:
When we say "after many iterations," it implies repeatedly applying the linear function to its previous output. Let's consider starting with a specific initial x-value and applying the function several times to see if the values converge towards a particular number.
For example, let's choose x = 0 as our initial value:
1. f1(0) = -0.5(0) + 2 = 2
- Applying the function to the initial value, we get 2 as the output.
Now, let's repeatedly apply the function:
f1(2) = -0.5(2) + 2 = 1
f1(1) = -0.5(1) + 2 = 1.5
f1(1.5) = -0.5(1.5) + 2 = 1.25
f1(1.25) = -0.5(1.25) + 2 = 1.375
...
As we continue to apply the function repeatedly, the values are approaching 1. This suggests that the function converges towards a particular number (in this case, 1) after many iterations.
Now, let's repeat this process with the second linear function:
2. f2(0) = -0.75(0) - 3 = -3
Repeating the function application:
f2(-3) = -0.75(-3) - 3 = -0.75
f2(-0.75) = -0.75(-0.75) - 3 ≈ -2.4375
f2(-2.4375) = -0.75(-2.4375) - 3 ≈ -3.0781
...
Here, the values are not converging towards a particular number but oscillating between different values as we continue to apply the function. In this case, there is no specific number to which the values are getting close.
Therefore, the behavior of the function's values after many iterations depends on the function's initial values and its properties (slope and y-intercept) within the given range.