Given F ( X ) = X 2 + 6 X − 27 X 2 − 6 X − 7 F(x)=\frac{x^2+6x-27}{x^2-6x-7} F ( X ) = X 2 − 6 X − 7 X 2 + 6 X − 27 ​ And G ( X ) = X − 3 X − 7 G(x)=\frac{x-3}{x-7} G ( X ) = X − 7 X − 3 ​ , Find ( F / G ) ( X (f / G)(x ( F / G ) ( X ] And Determine Any Domain Restrictions.Select The Correct Answer Below:A. ( F / G ) ( X ) = X − 9 X − 1 (f / G)(x)=\frac{x-9}{x-1} ( F / G ) ( X ) = X − 1 X − 9 ​ Where $x \neq -7, 1,

Introduction

In algebra, rational functions are a type of function that can be expressed as the ratio of two polynomials. Simplifying rational functions is an essential skill in mathematics, as it allows us to rewrite complex functions in a more manageable form. In this article, we will explore how to simplify the rational function $(f / g)(x)$, where $f(x)=\frac{x^2+6x-27}{x^2-6x-7}$ and $g(x)=\frac{x-3}{x-7}$.

Step 1: Factor the Numerator and Denominator of f(x)

To simplify the rational function $(f / g)(x)$, we first need to factor the numerator and denominator of $f(x)$. The numerator of $f(x)$ is $x^2+6x-27$, which can be factored as $(x+9)(x-3)$. The denominator of $f(x)$ is $x^2-6x-7$, which can be factored as $(x-7)(x+1)$.

f(x) = \frac{(x+9)(x-3)}{(x-7)(x+1)}

Step 2: Factor the Numerator and Denominator of g(x)

Next, we need to factor the numerator and denominator of $g(x)$. The numerator of $g(x)$ is $x-3$, which is already factored. The denominator of $g(x)$ is $x-7$, which is also already factored.

g(x) = \frac{x-3}{x-7}

Step 3: Simplify the Rational Function (f / g)(x)

Now that we have factored the numerator and denominator of $f(x)$ and $g(x)$, we can simplify the rational function $(f / g)(x)$. To do this, we need to divide the numerator of $f(x)$ by the numerator of $g(x)$, and then divide the denominator of $f(x)$ by the denominator of $g(x)$.

(f / g)(x) = \frac{\frac{(x+9)(x-3)}{(x-7)(x+1)}}{\frac{x-3}{x-7}}

Step 4: Cancel Common Factors

When we simplify the rational function $(f / g)(x)$, we can cancel common factors between the numerator and denominator. In this case, we can cancel the factor $(x-3)$ from the numerator and denominator.

(f / g)(x) = \frac{(x+9)}{(x+1)}

Step 5: Determine Domain Restrictions

Finally, we need to determine any domain restrictions for the simplified rational function $(f / g)(x)$. Since the denominator of $(f / g)(x)$ is $(x+1)$, we know that $x \neq -1$. Additionally, since the original function $f(x)$ had a denominator of $(x-7)(x+1)$, we also know that $x \neq 7$.

Conclusion

In conclusion, we have simplified the rational function $(f / g)(x)$ by factoring the numerator and denominator of $f(x)$ and $g(x)$, and then canceling common factors. We have also determined that the domain restrictions for $(f / g)(x)$ are $x \neq -1$ and $x \neq 7$.

Final Answer

The final answer is:

$(f / g)(x) = \frac{x-9}{x-1}$ where $x \neq -7, 1$

Introduction

In our previous article, we explored how to simplify the rational function $(f / g)(x)$, where $f(x)=\frac{x^2+6x-27}{x^2-6x-7}$ and $g(x)=\frac{x-3}{x-7}$. We factored the numerator and denominator of $f(x)$ and $g(x)$, and then canceled common factors to simplify the rational function. In this article, we will answer some common questions about simplifying rational functions.

Q: What is the first step in simplifying a rational function?

A: The first step in simplifying a rational function is to factor the numerator and denominator of the function. This will help you identify any common factors that can be canceled out.

Q: How do I factor the numerator and denominator of a rational function?

A: To factor the numerator and denominator of a rational function, you can use the following steps:

• Factor out any common factors from the numerator and denominator.
• Use the distributive property to expand the numerator and denominator.
• Look for any common factors that can be canceled out.

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

A: A rational function is a function that can be expressed as the ratio of two polynomials. A polynomial function, on the other hand, is a function that can be expressed as a sum of terms, where each term is a constant or a variable raised to a power.

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

A: To determine the domain restrictions of a rational function, you need to identify any values of x that would make the denominator of the function equal to zero. These values are called the domain restrictions.

Q: Can I cancel out a factor that is not common to both the numerator and denominator?

A: No, you cannot cancel out a factor that is not common to both the numerator and denominator. Canceling out a factor that is not common to both the numerator and denominator would be an error.

Q: What is the final answer to the problem of simplifying the rational function (f / g)(x)?

A: The final answer to the problem of simplifying the rational function (f / g)(x) is:

$(f / g)(x) = \frac{x-9}{x-1}$ where $x \neq -7, 1$

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

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

• Canceling out a factor that is not common to both the numerator and denominator.
• Failing to identify the domain restrictions of the function.
• Not factoring the numerator and denominator of the function.

Conclusion

In conclusion, simplifying rational functions is an essential skill in mathematics. By following the steps outlined in this article, you can simplify rational functions and determine their domain restrictions. Remember to avoid common mistakes, such as canceling out a factor that is not common to both the numerator and denominator.

Final Tips

• Always factor the numerator and denominator of a rational function before simplifying it.
• Identify the domain restrictions of a rational function before simplifying it.
• Avoid canceling out a factor that is not common to both the numerator and denominator.
• Practice simplifying rational functions to become more comfortable with the process.