# Answered Essay: Plot the function f(x) = (1- x) – 1 and the function x at a thousand points along the interval (-3 times 10^-15, +3 times 10 Plot the function f(x) = (1- x) – 1 and the function x at a thousand points along the interval (-3 times 10^-15, +3 times 10^-15). Here is the basic Python code to do this: import numpy as np from matplotlib import pyplot as plt # get 1000 points in the interval x = np.linspace (-3e-15, 3e-15, 1000) y_1 = x y_2 = (1 – x) – 1 plt.plot (x, y_1, ‘blue’) #plot y = x plt.plot (x, y2, ‘green’) #plot y = (1 – x) – 1 plt.show() #display the plot Note that (1 – x) – 1 = -x. Explain why the graph of x looks smooth but the graph of (1 – x) – 1 is jagged. Now plot the ratio ((1 – x) – 1)/x. Why does the error (the distance from the correct value of -1) get larger as x Rightarrow 0?

Python Code:

```import numpy as np
```
```from matplotlib import pyplot as plt

#get 1000 points in the interval
x = np.linspace(-3e-15, 3e-15, 1000)
y1 = x
y2 = (1-x)-1
plt.plot(x, y1, 'blue')
plt.plot(x, y2, 'green')

plt.show()
```

————————————————————————————————————————————————

1. Floating point operation need not be associative or commutative. So (1-x)-1 != (1-1)-x. Here (1-3e-15) = 1 (because of minimal resolution of floating points) leading to saturation of result. 2. ((1-x)-1) != -x. So   ((1-x)-1) /x != -1. Further when x is close to 0, ((1-x)-1)/x -> inf because of divide by 0 issue. Near x = 0, 0/0’s limit is not achieved.

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