determine the domain of the following graph:

Guri Bee

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20-10-2024

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Understanding the Domain of a Graph: A Visual Guide

Determining the domain of a graph is a fundamental concept in mathematics, especially when working with functions. The domain represents all possible input values (often represented by the x-axis) for which the function is defined.

Here's how to visually determine the domain of a graph:

1. Identify the x-axis: The horizontal axis on the graph represents the input values (x-values).
2. Look for any gaps or breaks: Pay close attention to where the graph exists and where it doesn't.
3. Determine the range of x-values covered:
   • If the graph extends infinitely to the left and right, the domain is all real numbers.
   • If the graph has a starting and ending point on the x-axis, the domain will be a specific interval.
   • If there are any gaps or breaks in the graph, the domain will exclude those x-values.

Example:

Let's say we have a graph of a function that looks like a straight line extending from x = -2 to x = 4.

• Step 1: We identify the x-axis.
• Step 2: The graph is continuous and doesn't have any breaks or gaps within the range of x-values.
• Step 3: The graph starts at x = -2 and ends at x = 4.

Therefore, the domain of this graph is -2 ≤ x ≤ 4.

Key Points to Remember:

• Open vs. Closed Intervals: A closed interval (represented by brackets []) includes the endpoint, while an open interval (represented by parentheses ()) excludes the endpoint.
• Vertical Asymptotes: If the graph approaches a vertical line without ever touching it (an asymptote), the domain excludes that x-value.
• Holes: If the graph has a "hole" at a specific x-value, the domain excludes that x-value.

Using Code to Visualize:

While we can determine the domain visually, using a graphing calculator or coding libraries like matplotlib in Python can make the process more precise and interactive. For example, you can plot the function and use the xlim function to set the limits of the x-axis to visually see the domain.

Example with matplotlib:

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-5, 5, 100)
y = x**2 

plt.plot(x, y)
plt.xlim(-5, 5)
plt.show()

This code will generate a graph of the function y = x² and set the x-axis limits to -5 and 5.

Understanding the domain of a graph is crucial for understanding the behavior of functions. By carefully analyzing the graph and using tools like graphing calculators or code, you can accurately determine the domain and better understand the limitations of the function.