What Is The Domain And Range Of The Function $f(x)=\frac{x+2}{x^2-x-12}$?Select Two Answer Choices: One For The Domain, And One For The Range.Domain:A. $(-\infty,-3) \cup(-3,4) \cup(4, \infty$\]B. $(-\infty, \infty$\]C.

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Introduction

When dealing 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) that the function can produce. In this article, we will explore the domain and range of the function f(x)=x+2x2−x−12f(x)=\frac{x+2}{x^2-x-12}.

What is the Domain of a Rational Function?

The domain of a rational function is the set of all real numbers except those that make the denominator equal to zero. To find the domain of the function f(x)=x+2x2−x−12f(x)=\frac{x+2}{x^2-x-12}, we need to find the values of x that make the denominator x2−x−12x^2-x-12 equal to zero.

Finding the Values of x that Make the Denominator Equal to Zero

To find the values of x that make the denominator equal to zero, we need to solve the equation x2−x−12=0x^2-x-12=0. We can factor the quadratic expression as (x−4)(x+3)=0(x-4)(x+3)=0. This gives us two possible values of x: x=4x=4 and x=−3x=-3.

Writing the Domain in Interval Notation

Since the values x=4x=4 and x=−3x=-3 make the denominator equal to zero, they are not included in the domain of the function. Therefore, the domain of the function f(x)=x+2x2−x−12f(x)=\frac{x+2}{x^2-x-12} is the set of all real numbers except x=4x=4 and x=−3x=-3. In interval notation, this can be written as (−∞,−3)∪(−3,4)∪(4,∞)(-\infty,-3) \cup(-3,4) \cup(4, \infty).

What is the Range of a Rational Function?

The range of a rational function is the set of all possible output values (y-values) that the function can produce. To find the range of the function f(x)=x+2x2−x−12f(x)=\frac{x+2}{x^2-x-12}, we need to consider the behavior of the function as x approaches positive and negative infinity.

Finding the Horizontal Asymptote

As x approaches positive and negative infinity, the function f(x)=x+2x2−x−12f(x)=\frac{x+2}{x^2-x-12} approaches a horizontal asymptote. To find the horizontal asymptote, we can divide the numerator and denominator by the highest power of x, which is x2x^2. This gives us xx2+2x21−xx2−12x2\frac{\frac{x}{x^2}+\frac{2}{x^2}}{1-\frac{x}{x^2}-\frac{12}{x^2}}. As x approaches positive and negative infinity, the terms xx2\frac{x}{x^2} and 2x2\frac{2}{x^2} approach zero, and the term 1−xx2−12x21-\frac{x}{x^2}-\frac{12}{x^2} approaches 1. Therefore, the horizontal asymptote is y=1y=1.

Writing the Range in Interval Notation

Since the horizontal asymptote is y=1y=1, the range of the function f(x)=x+2x2−x−12f(x)=\frac{x+2}{x^2-x-12} is the set of all real numbers except y=1y=1. However, since the function can approach y=1y=1 from both above and below, the range is actually the set of all real numbers except y=1y=1. In interval notation, this can be written as (−∞,1)∪(1,∞)(-\infty, 1) \cup (1, \infty).

Conclusion

In conclusion, the domain of the function f(x)=x+2x2−x−12f(x)=\frac{x+2}{x^2-x-12} is the set of all real numbers except x=4x=4 and x=−3x=-3, which can be written as (−∞,−3)∪(−3,4)∪(4,∞)(-\infty,-3) \cup(-3,4) \cup(4, \infty). The range of the function is the set of all real numbers except y=1y=1, which can be written as (−∞,1)∪(1,∞)(-\infty, 1) \cup (1, \infty).

Answer Choices

Based on our analysis, the correct answer choices are:

  • Domain: (−∞,−3)∪(−3,4)∪(4,∞)(-\infty,-3) \cup(-3,4) \cup(4, \infty)
  • Range: (−∞,1)∪(1,∞)(-\infty, 1) \cup (1, \infty)

Introduction

In our previous article, we explored the concept of domain and range of rational functions. We analyzed the function f(x)=x+2x2−x−12f(x)=\frac{x+2}{x^2-x-12} and found its domain and range. In this article, we will answer some frequently asked questions about the domain and range of rational functions.

Q: What is the domain of a rational function?

A: The domain of a rational function is the set of all real numbers except those that make the denominator equal to 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 equal to zero. You can do this by solving the equation ax2+bx+c=0ax^2+bx+c=0, where aa, bb, and cc are the coefficients of the quadratic expression in the denominator.

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) that the function can produce.

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

A: To find the range of a rational function, you need to consider the behavior of the function as x approaches positive and negative infinity. You can do this by finding the horizontal asymptote of the function.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that the function approaches as x approaches positive and negative infinity.

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

A: To find the horizontal asymptote of a rational function, you need to divide the numerator and denominator by the highest power of x. This will give you the equation of the horizontal asymptote.

Q: What is the difference between the domain and range of a rational function?

A: The domain of a rational function is the set of all real numbers except those that make the denominator equal to zero. The range of a rational function is the set of all possible output values (y-values) that the function can produce.

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

A: Yes, a rational function can have a domain of all real numbers if the denominator is never equal to zero.

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

A: No, a rational function cannot have a range of all real numbers. The range of a rational function is always a subset of the real numbers.

Q: How do I determine the domain and range of a rational function?

A: To determine the domain and range of a rational function, you need to analyze the function and consider the values of x that make the denominator equal to zero, as well as the behavior of the function as x approaches positive and negative infinity.

Conclusion

In conclusion, the domain and range of a rational function are important concepts that can help you understand the behavior of the function. By analyzing the function and considering the values of x that make the denominator equal to zero, as well as the behavior of the function as x approaches positive and negative infinity, you can determine the domain and range of a rational function.

Common Mistakes to Avoid

  • Not considering the values of x that make the denominator equal to zero when determining the domain of a rational function.
  • Not analyzing the behavior of the function as x approaches positive and negative infinity when determining the range of a rational function.
  • Not using the correct notation when writing the domain and range of a rational function.

Additional Resources

Practice Problems

  • Find the domain and range of the function f(x)=x+2x2−x−12f(x)=\frac{x+2}{x^2-x-12}.
  • Find the domain and range of the function f(x)=x−1x2+2x+1f(x)=\frac{x-1}{x^2+2x+1}.
  • Find the domain and range of the function f(x)=x+1x2−4x+4f(x)=\frac{x+1}{x^2-4x+4}.

Answer Key

  • The domain of the function f(x)=x+2x2−x−12f(x)=\frac{x+2}{x^2-x-12} is (−∞,−3)∪(−3,4)∪(4,∞)(-\infty,-3) \cup(-3,4) \cup(4, \infty).
  • The range of the function f(x)=x−1x2+2x+1f(x)=\frac{x-1}{x^2+2x+1} is (−∞,1)∪(1,∞)(-\infty, 1) \cup (1, \infty).
  • The domain of the function f(x)=x+1x2−4x+4f(x)=\frac{x+1}{x^2-4x+4} is (−∞,2)∪(2,∞)(-\infty, 2) \cup (2, \infty).
  • The range of the function f(x)=x+1x2−4x+4f(x)=\frac{x+1}{x^2-4x+4} is (−∞,1)∪(1,∞)(-\infty, 1) \cup (1, \infty).