Warm-up Questions

1. For f(x) =(lambda)x
when lambda = .1 or .3, f(x)will go to 0
when lambda = 2 or 5, f(x) will go to infinity
when lambda = 1, f(x) will always to equal to x(there is no change)
when x = 0, f(x) always is 0
Since lambda is the slope of the line, when lambda(the slope) is between 0 and 1, the behavior of the system tends to 0. When lambda(the slope) is greater than 1, the behavior of the system tends to infinity.

2. For f(x) =-(lambda)x

when lambda = -0.5 or -0.2, f(x)will go to 0
when lambda = -3 or -1.8, f(x) will go to negative infinity
when lambda = 1, f(x) will alternate between x and -x
when x = 0, f(x) always is 0
Since lambda is the slope of the line, when lambda(the slope) is between 0 and -1, the behavior of the system tends to 0. When lambda(the slope) is less than -1, the behavior of the system tends to negative infinity.

4. The long term behavior of the square root function tends towards 1 for all numbers greater than 0. The only values of x that do not change are 1 and 0.

The graph of the square root funtion crosses the line f(x) = x at the point (1, 1). It also crosses both f(x) = x and f(x) = -x at (0, 0). These are the points that do not change in the square root system.

5. The graph of the cosine function crosses the lines f(x) = x and f(x) = -x at (sqrt(2)/2, sqrt(2)/2) and (-sqrt(2)/2, sqrt(2)/2) respectively. These are the points that do not change in the cosine system.