When Graphing The Function F ( X ) = X 2 − 81 X 2 − 11 X + 18 F(x)=\frac{x^2-81}{x^2-11x+18} F ( X ) = X 2 − 11 X + 18 X 2 − 81 On Your Graphing Calculator, What Is The Most Appropriate Viewing Window For Determining The Domain And Range Of The Function?A. X Min ⁡ = − 10 , X Max ⁡ = 10 X_{\min} = -10, X_{\max} = 10 X M I N = − 10 , X M A X = 10 ;

Mar 10, 2025 by ADMIN 355 views

Understanding the Domain and Range of Rational Functions

When working with rational functions, it's essential to understand the concept of domain and range. 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). In this article, we'll explore how to determine the domain and range of the rational function $f(x)=\frac{x^2-81}{x^2-11x+18}$ using a graphing calculator.

Graphing Rational Functions

Rational functions are defined as the ratio of two polynomials. In this case, the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ is a rational function because it's the ratio of two polynomials. To graph this function on a graphing calculator, we need to enter the function in the calculator's input field and adjust the viewing window to get an accurate representation of the function.

Choosing the Right Viewing Window

When graphing a rational function, it's crucial to choose the right viewing window to determine the domain and range of the function. The viewing window is the rectangular area on the graphing calculator's screen that displays the graph of the function. The viewing window is defined by the following parameters:

$X_{\min}$ : The minimum x-value in the viewing window.
$X_{\max}$ : The maximum x-value in the viewing window.
$Y_{\min}$ : The minimum y-value in the viewing window.
$Y_{\max}$ : The maximum y-value in the viewing window.

To determine the domain and range of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ , we need to choose a viewing window that captures the entire graph of the function. The most appropriate viewing window for this function is one that includes the x-axis and the y-axis.

Determining the Domain

The domain of a rational function is the set of all possible input values (x-values) for which the function is defined. In the case of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ , the domain is all real numbers except for the values that make the denominator equal to zero.

To determine the domain of this function, we need to find the values of x that make the denominator equal to zero. We can do this by factoring the denominator and setting it equal to zero.

x^2 - 11x + 18 = (x - 3)(x - 6) = 0

Solving for x, we get:

x - 3 = 0 \quad \text{or} \quad x - 6 = 0

Solving for x, we get:

x = 3 \quad \text{or} \quad x = 6

Therefore, the domain of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ is all real numbers except for x = 3 and x = 6.

Determining the Range

The range of a rational function is the set of all possible output values (y-values) for which the function is defined. In the case of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ , the range is all real numbers except for the values that make the numerator equal to zero.

To determine the range of this function, we need to find the values of x that make the numerator equal to zero. We can do this by factoring the numerator and setting it equal to zero.

x^2 - 81 = (x - 9)(x + 9) = 0

Solving for x, we get:

x - 9 = 0 \quad \text{or} \quad x + 9 = 0

Solving for x, we get:

x = 9 \quad \text{or} \quad x = -9

Therefore, the range of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ is all real numbers except for y = 1 and y = -1.

Choosing the Right Viewing Window

A good starting point for the viewing window is to set $X_{\min}$ and $X_{\max}$ to -10 and 10, respectively. This will give us a good idea of the shape of the graph and the location of the x-intercepts.

X_{\min} = -10
X_{\max} = 10

We can then adjust the viewing window by changing the values of $X_{\min}$ and $X_{\max}$ to get a better view of the graph.

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 one that includes the x-axis and the y-axis. A good starting point for the viewing window is to set $X_{\min}$ and $X_{\max}$ to -10 and 10, respectively. This will give us a good idea of the shape of the graph and the location of the x-intercepts. By adjusting the viewing window, we can get a better view of the graph and determine the domain and range of the function.

References

[1] "Graphing Rational Functions" by Math Open Reference
[2] "Domain and Range of Rational Functions" by Purplemath
[3] "Graphing Calculators" by Texas Instruments

Additional Resources

[1] "Graphing Rational Functions" by Khan Academy
[2] "Domain and Range of Rational Functions" by Mathway
[3] "Graphing Calculators" by Casio
Frequently Asked Questions (FAQs) About Graphing Rational Functions

In this article, we'll answer some of the most frequently asked questions about graphing rational functions, including the domain and range of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ .

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

A: The domain of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ is all real numbers except for x = 3 and x = 6.

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

A: The range of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ is all real numbers except for y = 1 and y = -1.

Q: How do I choose the right viewing window for graphing the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ ?

A: To choose the right viewing window for graphing the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ , you should set $X_{\min}$ and $X_{\max}$ to -10 and 10, respectively. This will give you a good idea of the shape of the graph and the location of the x-intercepts.

Q: What is the significance of the x-intercepts in the graph of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ ?

A: The x-intercepts in the graph of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ represent the values of x that make the function equal to zero. These values are x = 3 and x = 6.

Q: How do I determine the y-intercept of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ ?

A: To determine the y-intercept of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ , you should set x = 0 and solve for y. This will give you the value of the y-intercept.

Q: What is the significance of the y-intercept in the graph of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ ?

A: The y-intercept in the graph of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ represents the value of y when x = 0. This value is y = 1.

Q: How do I graph the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ using a graphing calculator?

A: To graph the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ using a graphing calculator, you should enter the function in the calculator's input field and adjust the viewing window to get an accurate representation of the function.

Q: What are some common mistakes to avoid when graphing rational functions?

A: Some common mistakes to avoid when graphing rational functions include:

Not setting the viewing window correctly
Not factoring the numerator and denominator correctly
Not identifying the x-intercepts and y-intercepts correctly
Not using the correct graphing calculator settings

Conclusion

In conclusion, graphing rational functions can be a complex task, but with the right tools and techniques, you can get an accurate representation of the function. By following the steps outlined in this article, you can determine the domain and range of the function $f(x)=\frac{x^2-81}{x^2-11x+18}$ and graph the function using a graphing calculator.

References

[1] "Graphing Rational Functions" by Math Open Reference
[2] "Domain and Range of Rational Functions" by Purplemath
[3] "Graphing Calculators" by Texas Instruments

Additional Resources

[1] "Graphing Rational Functions" by Khan Academy
[2] "Domain and Range of Rational Functions" by Mathway
[3] "Graphing Calculators" by Casio