What Is The Domain Of $\left(\frac{f}{g}\right)(x$\]?Given:$\begin{array}{l} F(x) = 3x + 2 \\ G(x) = X^2 + 8x \end{array}$A. All Real Numbers Except 0B. All Real NumbersC. All Real Numbers Except -8 And 0D. All Real Numbers Except -8

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Introduction

When dealing with rational functions, it's essential to understand the concept of the 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 values of x that make the denominator equal to zero. In this article, we will explore the domain of the rational function $\left(\frac{f}{g}\right)(x)$, where $f(x) = 3x + 2$ and $g(x) = x^2 + 8x$.

Understanding the Domain of a Rational Function

A rational function is defined as the ratio of two polynomials. In this case, we have $f(x) = 3x + 2$ and $g(x) = x^2 + 8x$. The domain of a rational function is restricted by the values of x that make the denominator equal to zero. This is because division by zero is undefined in mathematics.

Finding the Domain of $\left(\frac{f}{g}\right)(x)$

To find the domain of $\left(\frac{f}{g}\right)(x)$, we need to find the values of x that make the denominator $g(x) = x^2 + 8x$ equal to zero. We can do this by solving the equation $x^2 + 8x = 0$.

Solving the Equation $x^2 + 8x = 0$

To solve the equation $x^2 + 8x = 0$, we can factor out the common term x:

$x(x + 8) = 0$

This tells us that either $x = 0$ or $x + 8 = 0$. Solving for x in the second equation, we get:

$x + 8 = 0 \Rightarrow x = -8$

Conclusion

Therefore, the values of x that make the denominator $g(x) = x^2 + 8x$ equal to zero are $x = 0$ and $x = -8$. This means that the domain of $\left(\frac{f}{g}\right)(x)$ is all real numbers except $x = 0$ and $x = -8$.

Final Answer

The final answer is $\boxed{C}$.

Discussion

The domain of a rational function is restricted by the values of x that make the denominator equal to zero. In this case, we found that the values of x that make the denominator $g(x) = x^2 + 8x$ equal to zero are $x = 0$ and $x = -8$. Therefore, the domain of $\left(\frac{f}{g}\right)(x)$ is all real numbers except $x = 0$ and $x = -8$.

Example

Let's consider an example to illustrate this concept. Suppose we have the rational function $\left(\frac{f}{g}\right)(x)$, where $f(x) = 3x + 2$ and $g(x) = x^2 + 8x$. We want to find the domain of this function.

Step 1: Find the values of x that make the denominator equal to zero

To find the domain of $\left(\frac{f}{g}\right)(x)$, we need to find the values of x that make the denominator $g(x) = x^2 + 8x$ equal to zero. We can do this by solving the equation $x^2 + 8x = 0$.

Step 2: Solve the equation $x^2 + 8x = 0$

To solve the equation $x^2 + 8x = 0$, we can factor out the common term x:

$x(x + 8) = 0$

This tells us that either $x = 0$ or $x + 8 = 0$. Solving for x in the second equation, we get:

$x + 8 = 0 \Rightarrow x = -8$

Step 3: Determine the domain of $\left(\frac{f}{g}\right)(x)$

Since the values of x that make the denominator $g(x) = x^2 + 8x$ equal to zero are $x = 0$ and $x = -8$, the domain of $\left(\frac{f}{g}\right)(x)$ is all real numbers except $x = 0$ and $x = -8$.

Step 4: Write the final answer

The final answer is $\boxed{C}$.

Conclusion

In conclusion, the domain of a rational function is restricted by the values of x that make the denominator equal to zero. In this case, we found that the values of x that make the denominator $g(x) = x^2 + 8x$ equal to zero are $x = 0$ and $x = -8$. Therefore, the domain of $\left(\frac{f}{g}\right)(x)$ is all real numbers except $x = 0$ and $x = -8$.

Final Thoughts

The concept of the domain of a rational function is essential in mathematics. It's crucial to understand how to find the domain of a rational function, as it can affect the validity of the function's output. In this article, we explored the domain of the rational function $\left(\frac{f}{g}\right)(x)$, where $f(x) = 3x + 2$ and $g(x) = x^2 + 8x$. We found that the domain of this function is all real numbers except $x = 0$ and $x = -8$.

Introduction

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

Q1: What is the domain of a rational function?

A1: 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 a rational function, the domain is restricted by the values of x that make the denominator equal to zero.

Q2: How do I find the domain of a rational function?

A2: 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 $g(x) = 0$, where $g(x)$ is the denominator of the rational function.

Q3: What happens if the denominator of a rational function is equal to zero?

A3: If the denominator of a rational function is equal to zero, the function is undefined at that point. This means that the value of the function is not defined for that particular input value.

Q4: Can a rational function have a domain of all real numbers?

A4: Yes, a rational function can have a domain of all real numbers if the denominator is never equal to zero for any real value of x.

Q5: How do I determine if a rational function has a domain of all real numbers?

A5: To determine if a rational function has a domain of all real numbers, you need to check if the denominator is ever equal to zero for any real value of x. If the denominator is never equal to zero, then the domain of the function is all real numbers.

Q6: What is the difference between the domain and the range of a function?

A6: The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-values) that the function can produce.

Q7: Can a rational function have a domain that is a subset of the real numbers?

A7: Yes, a rational function can have a domain that is a subset of the real numbers. For example, the domain of the function $f(x) = \frac{1}{x}$ is all real numbers except $x = 0$.

Q8: How do I graph a rational function?

A8: To graph a rational function, you need to first find the domain of the function. Then, you can use the graphing calculator or graph paper to plot the function.

Q9: Can a rational function have a domain that is a union of intervals?

A9: Yes, a rational function can have a domain that is a union of intervals. For example, the domain of the function $f(x) = \frac{1}{x}$ is the union of the intervals $(-\infty, 0)$ and $(0, \infty)$.

Q10: How do I find the domain of a rational function with a quadratic denominator?

A10: To find the domain of a rational function with a quadratic denominator, you need to solve the equation $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are the coefficients of the quadratic expression.

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. We have answered some frequently asked questions about the domain of a rational function and provided examples to illustrate the concepts.

Final Thoughts

The domain of a rational function is an essential concept in mathematics. It's crucial to understand how to find the domain of a rational function, as it can affect the validity of the function's output. In this article, we have provided answers to some frequently asked questions about the domain of a rational function and have provided examples to illustrate the concepts.

Example

Let's consider an example to illustrate the concept of the domain of a rational function. Suppose we have the rational function $f(x) = \frac{1}{x}$. We want to find the domain of this function.

Step 1: Find the values of x that make the denominator equal to zero

To find the domain of $f(x) = \frac{1}{x}$, we need to find the values of x that make the denominator equal to zero. We can do this by solving the equation $x = 0$.

Step 2: Solve the equation $x = 0$

To solve the equation $x = 0$, we can see that $x = 0$ is the only solution.

Step 3: Determine the domain of $f(x) = \frac{1}{x}$

Since the value of x that makes the denominator equal to zero is $x = 0$, the domain of $f(x) = \frac{1}{x}$ is all real numbers except $x = 0$.

Step 4: Write the final answer

The final answer is $\boxed{(-\infty, 0) \cup (0, \infty)}$.

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. We have answered some frequently asked questions about the domain of a rational function and provided examples to illustrate the concepts.