Find The Domain Of The Function:$ F(x) = \frac{x+4}{x^2-x-6} $Choose The Correct Domain Below:A. $ (-\infty, \infty) $ B. $ (-\infty, -3) \cup (-3, 2) \cup (2, \infty) $ C. $ (-\infty, -6) \cup (-6, -1) \cup (-1,

Introduction

When dealing with rational functions, it's essential to understand the concept of domain. 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 rational function, the domain is restricted by the presence of any denominators that may equal zero. In this article, we will explore how to find the domain of the function $f(x) = \frac{x+4}{x^2-x-6}$.

Understanding Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The general form of a rational function is:

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

where $p(x)$ and $q(x)$ are polynomials. The domain of a rational function is restricted by the values of $x$ that make the denominator $q(x)$ equal to zero.

Finding the Domain of a Rational Function

To find the domain of a rational function, we need to identify the values of $x$ that make the denominator equal to zero. We can do this by setting the denominator equal to zero and solving for $x$.

For the function $f(x) = \frac{x+4}{x^2-x-6}$, we need to find the values of $x$ that make the denominator $x^2-x-6$ equal to zero.

Step 1: Factor the Denominator

The first step is to factor the denominator $x^2-x-6$. We can factor it as:

$$x^2-x-6 = (x-3)(x+2)$$

Step 2: Set the Denominator Equal to Zero

Now that we have factored the denominator, we can set it equal to zero and solve for $x$.

$$(x-3)(x+2) = 0$$

Step 3: Solve for $x$

To solve for $x$, we can set each factor equal to zero and solve for $x$.

$$x-3 = 0 \Rightarrow x = 3$$

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

Step 4: Write the Domain

Now that we have found the values of $x$ that make the denominator equal to zero, we can write the domain of the function.

The domain of the function $f(x) = \frac{x+4}{x^2-x-6}$ is the set of all real numbers except $x = 3$ and $x = -2$.

Conclusion

In conclusion, the domain of a rational function is the set of all possible input values (x-values) for which the function is defined. To find the domain of a rational function, we need to identify the values of $x$ that make the denominator equal to zero and exclude those values from the domain. In this article, we have explored how to find the domain of the function $f(x) = \frac{x+4}{x^2-x-6}$.

Choosing the Correct Domain

Now that we have found the domain of the function, we can choose the correct domain from the options provided.

A. $(-\infty, \infty)$

B. $(-\infty, -3) \cup (-3, 2) \cup (2, \infty)$

C. $(-\infty, -6) \cup (-6, -1) \cup (-1, \infty)$

The correct domain is:

B. $(-\infty, -3) \cup (-3, 2) \cup (2, \infty)$

This is because the function is undefined at $x = 3$ and $x = -2$, which are the values that make the denominator equal to zero.

Final Answer

The final answer is:

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Introduction

In our previous article, we explored how to find the domain of a rational function. We discussed the concept of domain, how to identify the values of $x$ that make the denominator equal to zero, and how to write the domain of a function. In this article, we will answer some frequently asked questions about the domain of a rational function.

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 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 identify the values of $x$ that make the denominator equal to zero. You can do this by setting the denominator equal to zero and solving for $x$. Then, you can exclude those values from the domain.

Q: What happens if the denominator is a quadratic expression?

A: If the denominator is a quadratic expression, you can factor it to find the values of $x$ that make it equal to zero. For example, if the denominator is $x^2 - 4x + 4$, you can factor it as $(x - 2)^2$. Then, you can set each factor equal to zero and solve for $x$.

Q: Can the domain of a rational function be a single value?

A: Yes, the domain of a rational function can be a single value. For example, if the denominator is $x - 1$, the domain is $x = 1$. This is because the function is undefined at $x = 1$, which makes the denominator equal to zero.

Q: Can the domain of a rational function be an interval?

A: Yes, the domain of a rational function can be an interval. For example, if the denominator is $x^2 - 4$, the domain is $(-\infty, -2) \cup (2, \infty)$. This is because the function is undefined at $x = -2$ and $x = 2$, which make the denominator equal to zero.

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

A: To write the domain of a rational function, you need to exclude the values of $x$ that make the denominator equal to zero. You can do this by using interval notation. For example, if the domain is $(-\infty, -2) \cup (2, \infty)$, you can write it as $(-\infty, -2) \cup (2, \infty)$.

Q: Can the domain of a rational function be a union of intervals?

A: Yes, the domain of a rational function can be a union of intervals. For example, if the denominator is $x^2 - 4x + 4$, the domain is $(-\infty, -2) \cup (2, \infty)$. This is because the function is undefined at $x = -2$ and $x = 2$, which make the denominator equal to zero.

Q: How do I determine if a rational function is defined at a particular value of $x$?

A: To determine if a rational function is defined at a particular value of $x$, you need to check if the denominator is equal to zero at that value. If the denominator is equal to zero, the function is undefined at that value.

Conclusion

In conclusion, the domain of a rational function is the set of all possible input values (x-values) for which the function is defined. To find the domain of a rational function, you need to identify the values of $x$ that make the denominator equal to zero and exclude those values from the domain. We hope this Q&A guide has helped you understand the concept of domain and how to find it for a rational function.

Final Answer

The final answer is:

The domain of a rational function is the set of all possible input values (x-values) for which the function is defined. To find the domain of a rational function, you need to identify the values of $x$ that make the denominator equal to zero and exclude those values from the domain.