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?

1) After many iterations the function will become negative (at some moment)

2) The values are not getting close to a particular number; they will be decreasing indefinetly (toward - ∞)

Justification:

1) You can choose any value of the slope,m, in the interval (-1,0)

2) Use, for example, m = - 1/2

3) Since the slope intercept form of the linear equation is y = mx + b, you have your equation is:

y = (-1/2)x + b

4) That means that for any consecutive itereation the value of the function f, will decrease 1/2.

Suppose that you start with a value y = 1

Then, the next value will be y = 1 - 1/2 = 1/2

The next value will be y = 1/2 - 1/2 = 0

The next value will be y = 0 - 1/2 = -  1/2.

And so on.

The value of y will be decreasing with each iteration. The function f has not limit when x approaches bigger values, it will always become smaller and smalller.

Mathematically, it is said that the limit of function f(x) and x approaches zero is negative infinity.