If $f(x) = X + 7$ And $g(x) = \frac{1}{x - 13}$, What Is The Domain Of $(f \circ G)(x$\]?A. $\{x \mid X \neq 8\}$ B. $\{x \mid X = -8\}$ C. $\{x \mid X \neq -13\}$ D. $\{x \mid X \neq 13\}$

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Introduction

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When working with composite functions, it's essential to understand the concept of the domain. The domain of a composite function is the set of all possible input values for which the function is defined. In this article, we'll explore the domain of the composite function $(f \circ g)(x)$, where $f(x) = x + 7$ and $g(x) = \frac{1}{x - 13}$.

Understanding the Domain of a Function

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The domain of a function is the set of all possible input values for which the function is defined. In other words, it's the set of all possible values of $x$ for which the function $f(x)$ is defined. For example, the domain of the function $f(x) = \frac{1}{x}$ is all real numbers except $0$, because division by zero is undefined.

The Domain of $f(x)$

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The function $f(x) = x + 7$ is a simple linear function. Since it's a linear function, its domain is all real numbers. In other words, $f(x)$ is defined for all values of $x$.

The Domain of $g(x)$

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The function $g(x) = \frac{1}{x - 13}$ is a rational function. Since it's a rational function, its domain is all real numbers except the value that makes the denominator equal to zero. In this case, the denominator is $x - 13$, so the domain of $g(x)$ is all real numbers except $x = 13$.

The Composite Function $(f \circ g)(x)$

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The composite function $(f \circ g)(x)$ is defined as $f(g(x))$. In other words, we first apply the function $g(x)$ to the input value $x$, and then apply the function $f(x)$ to the result.

Finding the Domain of $(f \circ g)(x)$

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To find the domain of $(f \circ g)(x)$, we need to consider the restrictions on the domain of both $f(x)$ and $g(x)$. Since $f(x)$ is defined for all real numbers, the only restriction comes from the domain of $g(x)$, which is all real numbers except $x = 13$.

Solving for the Domain of $(f \circ g)(x)$

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To find the domain of $(f \circ g)(x)$, we need to solve the equation $x - 13 \neq 0$. This is because the denominator of the function $g(x)$ cannot be equal to zero. Solving for $x$, we get $x \neq 13$.

Conclusion

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In conclusion, the domain of the composite function $(f \circ g)(x)$ is all real numbers except $x = 13$. This is because the function $g(x)$ is undefined when the denominator is equal to zero, which occurs when $x = 13$.

Answer

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The correct answer is:

• D. $\{x \mid x \neq 13\}$

This is because the domain of $(f \circ g)(x)$ is all real numbers except $x = 13$.

Final Thoughts

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In this article, we explored the concept of the domain of a composite function. We found that the domain of $(f \circ g)(x)$ is all real numbers except $x = 13$. This is because the function $g(x)$ is undefined when the denominator is equal to zero, which occurs when $x = 13$. We hope this article has provided a clear understanding of the concept of the domain of a composite function.

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Introduction

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In our previous article, we explored the concept of the domain of a composite function. We found that the domain of $(f \circ g)(x)$ is all real numbers except $x = 13$. In this article, we'll provide a Q&A guide to help you better understand the concept of the domain of a composite function.

Q&A

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Q: What is the domain of a composite function?

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A: The domain of a composite function is the set of all possible input values for which the function is defined.

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

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A: To find the domain of a composite function, you need to consider the restrictions on the domain of both functions in the composition. You can do this by finding the values of $x$ that make the denominator of the function equal to zero.

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

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A: The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values of the function.

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

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A: To determine the domain of a rational function, you need to find the values of $x$ that make the denominator equal to zero. These values are not included in the domain of the function.

Q: Can the domain of a composite function be empty?

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A: Yes, the domain of a composite function can be empty. This occurs when the denominator of the function is equal to zero for all values of $x$.

Q: How do I find the domain of a composite function with multiple restrictions?

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A: To find the domain of a composite function with multiple restrictions, you need to consider all the restrictions on the domain of both functions in the composition. You can do this by finding the intersection of the domains of both functions.

Q: What is the significance of the domain of a composite function?

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A: The domain of a composite function is significant because it determines the set of all possible input values for which the function is defined. This is important because it helps us to understand the behavior of the function and to make predictions about its output.

Conclusion

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In conclusion, the domain of a composite function is the set of all possible input values for which the function is defined. To find the domain of a composite function, you need to consider the restrictions on the domain of both functions in the composition. We hope this Q&A guide has provided a clear understanding of the concept of the domain of a composite function.

Final Thoughts

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In this article, we provided a Q&A guide to help you better understand the concept of the domain of a composite function. We hope this guide has been helpful in clarifying any questions you may have had about the domain of a composite function.

Common Mistakes to Avoid

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• Not considering the restrictions on the domain of both functions in the composition
• Not finding the intersection of the domains of both functions
• Not considering the values of $x$ that make the denominator equal to zero

Tips and Tricks

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• Use a table to organize your work and keep track of the restrictions on the domain of both functions in the composition
• Use a Venn diagram to visualize the intersection of the domains of both functions
• Check your work by plugging in different values of $x$ to see if the function is defined

Practice Problems

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• Find the domain of the composite function $(f \circ g)(x)$, where $f(x) = x + 7$ and $g(x) = \frac{1}{x - 13}$
• Find the domain of the composite function $(f \circ g)(x)$, where $f(x) = \frac{1}{x}$ and $g(x) = x^2 + 1$
• Find the domain of the composite function $(f \circ g)(x)$, where $f(x) = x^2 + 1$ and $g(x) = \frac{1}{x - 2}$

Answer Key

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• The domain of the composite function $(f \circ g)(x)$, where $f(x) = x + 7$ and $g(x) = \frac{1}{x - 13}$ is all real numbers except $x = 13$
• The domain of the composite function $(f \circ g)(x)$, where $f(x) = \frac{1}{x}$ and $g(x) = x^2 + 1$ is all real numbers except $x = 0$
• The domain of the composite function $(f \circ g)(x)$, where $f(x) = x^2 + 1$ and $g(x) = \frac{1}{x - 2}$ is all real numbers except $x = 2$