Identify The Characteristics Of The Function Below.$\[ F(x)=\frac{x^2+4x-12}{x^2-x-2} \\]- X-intercept: \[$(A, 0)\$\] Where \[$A = \, \square \$\]- Y-intercept: \[$(0, B)\$\] Where \[$B = \, \square \$\]-

Introduction

In mathematics, rational functions are a type of function that can be expressed as the ratio of two polynomials. They are an essential concept in algebra and are used to model various real-world phenomena. In this article, we will analyze the characteristics of a given rational function, including its x-intercept and y-intercept.

The Function

The given rational function is:

${ f(x) = \frac{x^2 + 4x - 12}{x^2 - x - 2} }$

To analyze the characteristics of this function, we need to factor the numerator and denominator.

Factoring the Numerator and Denominator

The numerator can be factored as:

${ x^2 + 4x - 12 = (x + 6)(x - 2) }$

The denominator can be factored as:

${ x^2 - x - 2 = (x - 2)(x + 1) }$

Simplifying the Function

Now that we have factored the numerator and denominator, we can simplify the function by canceling out any common factors.

${ f(x) = \frac{(x + 6)(x - 2)}{(x - 2)(x + 1)} }$

We can cancel out the common factor (x - 2) from the numerator and denominator:

${ f(x) = \frac{x + 6}{x + 1} }$

Finding the X-Intercept

The x-intercept is the point where the function intersects the x-axis, i.e., where y = 0. To find the x-intercept, we set f(x) = 0 and solve for x.

${ \frac{x + 6}{x + 1} = 0 }$

Since the numerator is equal to zero, we can set x + 6 = 0 and solve for x:

${ x + 6 = 0 }$ ${ x = -6 }$

Therefore, the x-intercept is (-6, 0).

Finding the Y-Intercept

The y-intercept is the point where the function intersects the y-axis, i.e., where x = 0. To find the y-intercept, we substitute x = 0 into the function and solve for y.

${ f(0) = \frac{0 + 6}{0 + 1} }$ ${ f(0) = 6 }$

Therefore, the y-intercept is (0, 6).

Conclusion

In this article, we analyzed the characteristics of a given rational function, including its x-intercept and y-intercept. We factored the numerator and denominator, simplified the function, and found the x-intercept and y-intercept. The x-intercept is (-6, 0) and the y-intercept is (0, 6). This analysis provides valuable insights into the behavior of the function and can be used to model various real-world phenomena.

Key Takeaways

• Rational functions can be expressed as the ratio of two polynomials.
• Factoring the numerator and denominator can simplify the function.
• The x-intercept is the point where the function intersects the x-axis, i.e., where y = 0.
• The y-intercept is the point where the function intersects the y-axis, i.e., where x = 0.

Further Reading

For more information on rational functions and their characteristics, we recommend the following resources:

• Khan Academy: Rational Functions
• Mathway: Rational Functions
• Wolfram Alpha: Rational Functions

Introduction

In our previous article, we analyzed the characteristics of a given rational function, including its x-intercept and y-intercept. In this article, we will answer some frequently asked questions about rational functions.

Q: What is a rational function?

A rational function is a type of function that can be expressed as the ratio of two polynomials. It is a function that can be written in the form:

${ f(x) = \frac{p(x)}{q(x)} }$

where p(x) and q(x) are polynomials.

Q: What are the characteristics of a rational function?

The characteristics of a rational function include:

• Domain: The set of all possible input values (x) for which the function is defined.
• Range: The set of all possible output values (y) for which the function is defined.
• X-intercept: The point where the function intersects the x-axis, i.e., where y = 0.
• Y-intercept: The point where the function intersects the y-axis, i.e., where x = 0.

Q: How do I find the x-intercept of a rational function?

To find the x-intercept of a rational function, you need to set the function equal to zero and solve for x. This can be done by setting the numerator equal to zero and solving for x.

Q: How do I find the y-intercept of a rational function?

To find the y-intercept of a rational function, you need to substitute x = 0 into the function and solve for y.

Q: What is the difference between a rational function and a polynomial function?

A rational function is a function that can be expressed as the ratio of two polynomials, while a polynomial function is a function that can be expressed as a sum of terms, each of which is a constant or a product of constants and variables.

Q: Can a rational function have a hole in its graph?

Yes, a rational function can have a hole in its graph if there is a common factor in the numerator and denominator that is canceled out.

Q: How do I graph a rational function?

To graph a rational function, you need to:

1. Find the x-intercepts: Set the function equal to zero and solve for x.
2. Find the y-intercept: Substitute x = 0 into the function and solve for y.
3. Find the asymptotes: Find the vertical and horizontal asymptotes of the function.
4. Plot the graph: Use the x-intercepts, y-intercept, and asymptotes to plot the graph of the function.

Conclusion

In this article, we answered some frequently asked questions about rational functions. We covered topics such as the definition of a rational function, its characteristics, and how to find the x-intercept and y-intercept. We also discussed the difference between a rational function and a polynomial function, and how to graph a rational function.

Key Takeaways

• A rational function is a function that can be expressed as the ratio of two polynomials.
• The characteristics of a rational function include its domain, range, x-intercept, and y-intercept.
• To find the x-intercept of a rational function, set the function equal to zero and solve for x.
• To find the y-intercept of a rational function, substitute x = 0 into the function and solve for y.
• A rational function can have a hole in its graph if there is a common factor in the numerator and denominator that is canceled out.

Further Reading

For more information on rational functions and their characteristics, we recommend the following resources:

• Khan Academy: Rational Functions
• Mathway: Rational Functions
• Wolfram Alpha: Rational Functions

By understanding the characteristics of rational functions, we can better model and analyze real-world phenomena, leading to new insights and discoveries.