Find The Domain Of The Function. F ( X ) = 1 6 X + 1 − 2 F(x) = \frac{1}{\frac{6}{x+1} - 2} F ( X ) = X + 1 6 ​ − 2 1 ​ The Domain Of F ( X F(x F ( X ] Is □ \square □ .(Type Your Answer In Interval Notation.)

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Introduction

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In mathematics, the domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x for which the function is valid. In this article, we will find the domain of the function $f(x) = \frac{1}{\frac{6}{x+1} - 2}$.

Understanding the Function

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The given function is a rational function, which means it is a function that can be expressed as the ratio of two polynomials. In this case, the function is $f(x) = \frac{1}{\frac{6}{x+1} - 2}$. To find the domain of this function, we need to consider the values of x for which the function is defined.

Identifying the Restrictions

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The function $f(x) = \frac{1}{\frac{6}{x+1} - 2}$ has two restrictions:

1. The denominator of the function cannot be equal to zero.
2. The expression inside the inner parentheses cannot be equal to zero.

Finding the Restrictions

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Let's analyze the function to find the restrictions.

Restriction 1: Denominator Cannot be Zero

The denominator of the function is $\frac{6}{x+1} - 2$. To find the values of x for which the denominator is not equal to zero, we need to set the denominator equal to zero and solve for x.

$\frac{6}{x+1} - 2 = 0$

Solving for x, we get:

$\frac{6}{x+1} = 2$

$6 = 2(x+1)$

$6 = 2x + 2$

$4 = 2x$

$2 = x$

So, the denominator is equal to zero when x = 2.

Restriction 2: Expression Inside Inner Parentheses Cannot be Zero

The expression inside the inner parentheses is $x+1$. To find the values of x for which this expression is not equal to zero, we need to set the expression equal to zero and solve for x.

$x+1 = 0$

Solving for x, we get:

$x = -1$

So, the expression inside the inner parentheses is equal to zero when x = -1.

Finding the Domain

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Now that we have identified the restrictions, we can find the domain of the function.

The domain of the function is all real numbers except for the values of x that make the denominator equal to zero or the expression inside the inner parentheses equal to zero.

In interval notation, the domain of the function is:

$(-\infty, -1) \cup (-1, 2) \cup (2, \infty)$

Conclusion

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In this article, we found the domain of the function $f(x) = \frac{1}{\frac{6}{x+1} - 2}$. The domain of the function is all real numbers except for the values of x that make the denominator equal to zero or the expression inside the inner parentheses equal to zero. The domain of the function is $(-\infty, -1) \cup (-1, 2) \cup (2, \infty)$.

Final Answer

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The final answer is $\boxed{(-\infty, -1) \cup (-1, 2) \cup (2, \infty)}$.

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Introduction

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In our previous article, we found the domain of the function $f(x) = \frac{1}{\frac{6}{x+1} - 2}$. In this article, we will answer some frequently asked questions related to the domain of the function.

Q&A

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Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x for which the function is valid.

Q: Why is it important to find the domain of a function?

A: Finding the domain of a function is important because it helps us to determine the values of x for which the function is defined. This is crucial in mathematics, as it allows us to perform calculations and make conclusions about the function.

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

A: To find the domain of a rational function, you need to consider the values of x for which the denominator is not equal to zero. You also need to consider the values of x for which the expression inside the inner parentheses is not equal to zero.

Q: What are the restrictions on the domain of a rational function?

A: The restrictions on the domain of a rational function are:

1. The denominator of the function cannot be equal to zero.
2. The expression inside the inner parentheses cannot be equal to zero.

Q: How do I find the values of x that make the denominator equal to zero?

A: To find the values of x that make the denominator equal to zero, you need to set the denominator equal to zero and solve for x.

Q: How do I find the values of x that make the expression inside the inner parentheses equal to zero?

A: To find the values of x that make the expression inside the inner parentheses equal to zero, you need to set the expression equal to zero and solve for x.

Q: What is the domain of the function $f(x) = \frac{1}{\frac{6}{x+1} - 2}$?

A: The domain of the function $f(x) = \frac{1}{\frac{6}{x+1} - 2}$ is $(-\infty, -1) \cup (-1, 2) \cup (2, \infty)$.

Q: Why is the domain of the function $f(x) = \frac{1}{\frac{6}{x+1} - 2}$ $(-\infty, -1) \cup (-1, 2) \cup (2, \infty)$?

A: The domain of the function $f(x) = \frac{1}{\frac{6}{x+1} - 2}$ is $(-\infty, -1) \cup (-1, 2) \cup (2, \infty)$ because the denominator of the function is equal to zero when x = 2, and the expression inside the inner parentheses is equal to zero when x = -1.

Conclusion

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In this article, we answered some frequently asked questions related to the domain of the function $f(x) = \frac{1}{\frac{6}{x+1} - 2}$. We hope that this article has helped you to understand the concept of the domain of a function and how to find it.

Final Answer

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The final answer is $\boxed{(-\infty, -1) \cup (-1, 2) \cup (2, \infty)}$.