Consider The Set Of Points \[$\left\{(x, F(x)) \left\lvert\, F(x)=\frac{1}{3} X+4\right. ; 0 \leq X \leq 6\right\}\$\]. Identify The Domain And Range Of This Function.

Introduction

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). When dealing with functions, it's essential to understand the domain and range, as they provide valuable information about the behavior of the function. In this article, we'll explore the domain and range of a linear function, specifically the set of points defined by the equation $f(x) = \frac{1}{3}x + 4$ for $0 \leq x \leq 6$.

What is a Linear Function?

A linear function is a function that can be written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope represents the rate of change of the function, while the y-intercept represents the point where the function intersects the y-axis. In our case, the linear function is defined by $f(x) = \frac{1}{3}x + 4$, where the slope is $\frac{1}{3}$ and the y-intercept is $4$.

Domain of the Function

The domain of a function is the set of all possible input values for which the function is defined. In other words, it's the set of all possible x-values that the function can take. For our linear function, the domain is defined as $0 \leq x \leq 6$, which means that the function is only defined for x-values between $0$ and $6$, inclusive.

Range of the Function

The range of a function is the set of all possible output values for which the function is defined. In other words, it's the set of all possible y-values that the function can take. To find the range of our linear function, we need to consider the minimum and maximum values of the function within the given domain.

Finding the Minimum and Maximum Values

To find the minimum and maximum values of the function, we can use the fact that the function is linear and the domain is a closed interval. The minimum value of the function occurs at the smallest x-value in the domain, which is $x = 0$. Plugging this value into the function, we get:

$f(0) = \frac{1}{3}(0) + 4 = 4$

The maximum value of the function occurs at the largest x-value in the domain, which is $x = 6$. Plugging this value into the function, we get:

$f(6) = \frac{1}{3}(6) + 4 = 6 + 4 = 10$

Conclusion

In conclusion, the domain of the function is $0 \leq x \leq 6$, and the range is $4 \leq f(x) \leq 10$. This means that the function takes on all values between $4$ and $10$ inclusive, and the function is only defined for x-values between $0$ and $6$, inclusive.

Visualizing the Function

To visualize the function, we can plot the points $(x, f(x))$ for $0 \leq x \leq 6$. This will give us a straight line that passes through the points $(0, 4)$ and $(6, 10)$. The line will have a slope of $\frac{1}{3}$ and a y-intercept of $4$.

Code to Visualize the Function

If you want to visualize the function using Python, you can use the following code:

import numpy as np
import matplotlib.pyplot as plt

# Define the function
def f(x):
    return (1/3)*x + 4

# Generate x-values
x = np.linspace(0, 6, 100)

# Generate y-values
y = f(x)

# Plot the points
plt.plot(x, y)

# Set the title and labels
plt.title('Linear Function')
plt.xlabel('x')
plt.ylabel('f(x)')

# Show the plot
plt.show()

This code will generate a plot of the function, which will help you visualize the domain and range of the function.

Conclusion

Frequently Asked Questions

In this article, we'll answer some frequently asked questions about the domain and range of a linear function.

Q: What is the domain of a linear function?

A: The domain of a linear function is the set of all possible input values for which the function is defined. In other words, it's the set of all possible x-values that the function can take.

Q: How do I find the domain of a linear function?

A: To find the domain of a linear function, you need to consider the restrictions on the input values. For example, if the function is defined as $f(x) = \frac{1}{3}x + 4$ for $0 \leq x \leq 6$, then the domain is $0 \leq x \leq 6$.

Q: What is the range of a linear function?

A: The range of a linear function is the set of all possible output values for which the function is defined. In other words, it's the set of all possible y-values that the function can take.

Q: How do I find the range of a linear function?

A: To find the range of a linear function, you need to consider the minimum and maximum values of the function within the given domain. For example, if the function is defined as $f(x) = \frac{1}{3}x + 4$ for $0 \leq x \leq 6$, then the minimum value of the function is $f(0) = 4$ and the maximum value of the function is $f(6) = 10$.

Q: Can a linear function have a domain of all real numbers?

A: Yes, a linear function can have a domain of all real numbers. For example, the function $f(x) = x + 4$ has a domain of all real numbers.

Q: Can a linear function have a range of all real numbers?

A: No, a linear function cannot have a range of all real numbers. The range of a linear function is always a line segment or a ray, and it cannot cover all real numbers.

Q: How do I determine if a linear function is increasing or decreasing?

A: To determine if a linear function is increasing or decreasing, you need to consider the slope of the function. If the slope is positive, then the function is increasing. If the slope is negative, then the function is decreasing.

Q: Can a linear function have a horizontal asymptote?

A: No, a linear function cannot have a horizontal asymptote. The horizontal asymptote of a function is a horizontal line that the function approaches as the input values approach infinity or negative infinity. Since a linear function is a straight line, it cannot approach a horizontal line as the input values approach infinity or negative infinity.

Q: Can a linear function have a vertical asymptote?

A: No, a linear function cannot have a vertical asymptote. The vertical asymptote of a function is a vertical line that the function approaches as the input values approach a certain value. Since a linear function is a straight line, it cannot approach a vertical line as the input values approach a certain value.

Conclusion

In conclusion, the domain and range of a linear function are essential concepts in mathematics. By understanding the domain and range of a linear function, you can gain valuable insights into the behavior of the function and make predictions about its output values. We hope this article has helped you understand the domain and range of a linear function and has answered some of your frequently asked questions.