Determining the Domain and Range of a Rational Function

Oct 13, 2025 by windailys.com 24 views

Determining the Domain and Range of a Rational Function

When graphing the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ on your graphing calculator, it is essential to choose the most appropriate viewing window to accurately determine the domain and range of the function. The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values).

Understanding the Function

The given function is a rational function, which means it is the ratio of two polynomials. The numerator is $x^2-81$ , and the denominator is $x^2-11x+18$ . To determine the domain and range of this function, we need to consider the values of x that make the denominator zero, as division by zero is undefined.

Finding the Domain

To find the domain of the function, we need to find the values of x that make the denominator zero. We can do this by factoring the denominator:

$x^2-11x+18 = (x-9)(x-2)$

Now, we can set each factor equal to zero and solve for x:

$x-9 = 0 \Rightarrow x = 9$

$x-2 = 0 \Rightarrow x = 2$

Therefore, the values of x that make the denominator zero are x = 2 and x = 9. These values are not in the domain of the function.

Choosing the Viewing Window

To determine the domain and range of the function, we need to choose a viewing window that includes the points x = 2 and x = 9. We also want to make sure that the viewing window is large enough to capture the behavior of the function near these points.

Option A: Xmin: -10, Xmax: 10, Ymin: -10, Ymax: 10

This viewing window is too large and includes many points that are not relevant to the function. It will also make it difficult to see the behavior of the function near the points x = 2 and x = 9.

Option B: Xmin: -5, Xmax: 5, Ymin: -10, Ymax: 10

This viewing window is more suitable for determining the domain and range of the function. It includes the points x = 2 and x = 9 and provides a clear view of the behavior of the function near these points.

Option C: Xmin: -1, Xmax: 1, Ymin: -10, Ymax: 10

This viewing window is even more focused and provides a clear view of the behavior of the function near the points x = 2 and x = 9. It is the most appropriate viewing window for determining the domain and range of the function.

Conclusion

In conclusion, the most appropriate viewing window for determining the domain and range of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ is Xmin: -1, Xmax: 1, Ymin: -10, Ymax: 10. This viewing window includes the points x = 2 and x = 9 and provides a clear view of the behavior of the function near these points.

Domain and Range

The domain of the function is all real numbers except x = 2 and x = 9. The range of the function is all real numbers except y = 0.

Graphing the Function

To graph the function, we can use the viewing window Xmin: -1, Xmax: 1, Ymin: -10, Ymax: 10. We can also use the graphing calculator's built-in functions to graph the function and determine its domain and range.

Final Answer

The final answer is Xmin: -1, Xmax: 1, Ymin: -10, Ymax: 10.
Q&A: Determining the Domain and Range of a Rational Function

In the previous article, we discussed how to determine the domain and range of a rational function using a graphing calculator. Here are some frequently asked questions and answers to help you better understand the concept.

Q: What is the domain of a rational function?

A: The domain of a rational function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of all real numbers except those that make the denominator zero.

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

A: To find the domain of a rational function, you need to find the values of x that make the denominator zero. You can do this by factoring the denominator and setting each factor equal to zero.

Q: What is the range of a rational function?

A: The range of a rational function is the set of all possible output values (y-values) for which the function is defined. In other words, it is the set of all real numbers except those that are not in the range of the numerator.

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

A: To find the range of a rational function, you need to find the values of y that are not in the range of the numerator. You can do this by graphing the function and observing the behavior of the graph.

Q: What is the most appropriate viewing window for determining the domain and range of a rational function?

A: The most appropriate viewing window for determining the domain and range of a rational function is one that includes the points that make the denominator zero and provides a clear view of the behavior of the function near these points.

Q: How do I choose the most appropriate viewing window?

A: To choose the most appropriate viewing window, you need to consider the values of x that make the denominator zero and the behavior of the function near these points. You can use the graphing calculator's built-in functions to graph the function and determine the most appropriate viewing window.

Q: What is the final answer for determining the domain and range of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ ?

A: The final answer is Xmin: -1, Xmax: 1, Ymin: -10, Ymax: 10.

Q: How do I graph the function using the graphing calculator?

A: To graph the function using the graphing calculator, you need to enter the function into the calculator and use the graphing function to graph the function. You can also use the calculator's built-in functions to graph the function and determine its domain and range.

Q: What are some common mistakes to avoid when determining the domain and range of a rational function?

A: Some common mistakes to avoid when determining the domain and range of a rational function include:

Not factoring the denominator correctly
Not setting each factor equal to zero
Not considering the behavior of the function near the points that make the denominator zero
Not choosing the most appropriate viewing window

Conclusion

In conclusion, determining the domain and range of a rational function is an important concept in mathematics. By following the steps outlined in this article and avoiding common mistakes, you can accurately determine the domain and range of a rational function using a graphing calculator.

Additional Resources

For more information on determining the domain and range of a rational function, you can consult the following resources:

Graphing calculator user manual
Mathematics textbook
Online tutorials and videos

Final Answer

The final answer is Xmin: -1, Xmax: 1, Ymin: -10, Ymax: 10.