If $f(x)=1032(x-3)$, Then Find The Domain, Range, Intercepts, And Sketch The Graph.

Mar 1, 2025 by ADMIN 86 views

Introduction

In this article, we will explore the properties of a linear function, specifically the domain, range, intercepts, and graph of the function $f(x)=1032(x-3)$. Understanding these properties is essential in mathematics, as it allows us to visualize and analyze the behavior of functions.

Domain of the Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of a linear function like $f(x)=1032(x-3)$, the domain is all real numbers, denoted as $(-\infty, \infty)$. This is because the function is defined for any value of x, and there are no restrictions on the input values.

Range of the Function

The range of a function is the set of all possible output values (y-values) for which the function is defined. To find the range of the function $f(x)=1032(x-3)$, we need to consider the behavior of the function as x approaches positive and negative infinity. As x approaches positive infinity, f(x) also approaches positive infinity. Similarly, as x approaches negative infinity, f(x) approaches negative infinity. This indicates that the range of the function is all real numbers, denoted as $(-\infty, \infty)$.

Intercepts of the Function

The intercepts of a function are the points where the function intersects the x-axis and y-axis. To find the x-intercept, we set f(x) = 0 and solve for x. In this case, we have:

1032(x-3) = 0

Solving for x, we get:

x = 3

This means that the x-intercept is at the point (3, 0).

To find the y-intercept, we set x = 0 and solve for f(x). In this case, we have:

f(0) = 1032(0-3)

Simplifying, we get:

f(0) = -3096

This means that the y-intercept is at the point (0, -3096).

Sketching the Graph

To sketch the graph of the function $f(x)=1032(x-3)$, we can use the intercepts and the behavior of the function as x approaches positive and negative infinity. The graph will be a straight line that passes through the x-intercept (3, 0) and the y-intercept (0, -3096). As x approaches positive infinity, f(x) approaches positive infinity, and as x approaches negative infinity, f(x) approaches negative infinity.

Conclusion

In this article, we have explored the properties of the linear function $f(x)=1032(x-3)$. We have found that the domain and range of the function are all real numbers, and we have identified the x-intercept and y-intercept of the function. We have also sketched the graph of the function, which is a straight line that passes through the intercepts and approaches positive and negative infinity as x approaches positive and negative infinity.

Final Thoughts

Understanding the properties of linear functions is essential in mathematics, as it allows us to analyze and visualize the behavior of functions. By exploring the domain, range, intercepts, and graph of the function $f(x)=1032(x-3)$, we have gained a deeper understanding of the properties of linear functions and how they can be used to model real-world phenomena.

References

[1] "Linear Functions" by Khan Academy
[2] "Domain and Range" by Math Open Reference
[3] "Intercepts" by Purplemath

Additional Resources

[1] "Graphing Linear Functions" by Mathway
[2] "Linear Function Properties" by IXL
[3] "Mathematics for Calculus" by MIT OpenCourseWare

Introduction

In our previous article, we explored the properties of the linear function $f(x)=1032(x-3)$. We found that the domain and range of the function are all real numbers, and we identified the x-intercept and y-intercept of the function. We also sketched the graph of the function, which is a straight line that passes through the intercepts and approaches positive and negative infinity as x approaches positive and negative infinity.

Q&A

Q: What is the domain of the function $f(x)=1032(x-3)$?

A: The domain of the function is all real numbers, denoted as $(-\infty, \infty)$. This is because the function is defined for any value of x, and there are no restrictions on the input values.

Q: What is the range of the function $f(x)=1032(x-3)$?

A: The range of the function is all real numbers, denoted as $(-\infty, \infty)$. This is because the function approaches positive and negative infinity as x approaches positive and negative infinity.

Q: What are the intercepts of the function $f(x)=1032(x-3)$?

A: The x-intercept is at the point (3, 0), and the y-intercept is at the point (0, -3096).

Q: How do you sketch the graph of the function $f(x)=1032(x-3)$?

A: To sketch the graph, you can use the intercepts and the behavior of the function as x approaches positive and negative infinity. The graph will be a straight line that passes through the intercepts and approaches positive and negative infinity as x approaches positive and negative infinity.

Q: What is the equation of the line that passes through the intercepts of the function $f(x)=1032(x-3)$?

A: The equation of the line is $y = 1032(x-3)$.

Q: What is the slope of the line that passes through the intercepts of the function $f(x)=1032(x-3)$?

A: The slope of the line is 1032.

Q: What is the y-intercept of the line that passes through the intercepts of the function $f(x)=1032(x-3)$?

A: The y-intercept is -3096.

Q: How do you find the equation of a line that passes through two points?

A: To find the equation of a line that passes through two points, you can use the slope-intercept form of a line, which is $y = mx + b$, where m is the slope and b is the y-intercept. You can find the slope by using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, and then use one of the points to find the y-intercept.

Conclusion

In this Q&A article, we have answered some common questions about the linear function $f(x)=1032(x-3)$. We have discussed the domain and range of the function, the intercepts of the function, and how to sketch the graph of the function. We have also provided some additional information about how to find the equation of a line that passes through two points.