asap needed in 30 mins (algebra help. graph)

Question

Sketch the graph of a function f with the following properties: - f is not continuous when x=2 but limx→2​f(x) exists. - f has a jump discontinuity when x=−1. - f is continuous at all points other than where x=2 or x=−1.

Answer 1

f(x) = x, if x ≤ -1

x + 1, if -1 < x < 2

5, if x = 2

-x + 5, if x > 2

Answer 2

Python will provide you with a fast way of answer this:

import matplotlib.pyplot as plt

import numpy as np

# Generate a range of x values

x = np.linspace(-3, 3, 400)

# Define f(x) with the described properties

def f(x):

y = np.zeros_like(x)

for i, val in enumerate(x):

if val < -1:

y[i] = val + 1 # Example linear function for x < -1

elif val == -1:

y[i] = np.nan # Jump discontinuity at x = -1

elif -1 < val < 2:

y[i] = val + 1 # Example linear function for -1 < x < 2

elif val == 2:

y[i] = np.nan # Remove point at x = 2

else:

y[i] = val + 1.5 # Example linear function for x > 2

return y

# Generate y values

y = f(x)

# Plotting the function

plt.plot(x, y, label=r'$f(x)$')

# Highlighting the discontinuities with dots

plt.plot(-1, f(np.array([-1.1])).item(), 'bo') # Left dot at x = -1

plt.plot(-1, f(np.array([-0.9])).item(), 'ro') # Right dot at x = -1

plt.plot(2, f(np.array([2.1])).item(), 'bo') # Dot at x = 2

# Adding labels and title

plt.xlabel(r'$x$')

plt.ylabel(r'$f(x)$')

plt.title(r'Graph of $f(x)$')

plt.legend()

plt.grid(True)

plt.axvline(x=-1, color='black', linestyle='--', linewidth=0.5)

plt.axvline(x=2, color='black', linestyle='--', linewidth=0.5)

plt.axhline(0, color='black',linewidth=0.5)

plt.axvline(0, color='black',linewidth=0.5)

# Show the plot

plt.show()

EXPLANATION

1. f(x): Define the function based on the discontinuity conditions specified. Utilize np.nan for the discontinuities to leave them blank on the graph.
2. plt.plot(x, y, label=r'$f(x)$'): Graph the function.
3. plt.plot(-1, f(np.array([-1.1])).item(), 'bo') etc.: Mark the discontinuity points with dots.