If F ( X ) = X + 7 F(x) = X + 7 F ( X ) = X + 7 And G ( X ) = 1 X − 13 G(x) = \frac{1}{x-13} G ( X ) = X − 13 1 ​ , What Is The Domain Of ( F ∘ G ) ( X (f \circ G)(x ( F ∘ G ) ( X ]?A. { X ∣ X ≠ 6 } \{x \mid X \neq 6\} { X ∣ X  = 6 } B. { X ∣ X = − 6 } \{x \mid X = -6\} { X ∣ X = − 6 } C. { X ∣ X ≠ − 13 } \{x \mid X \neq -13\} { X ∣ X  = − 13 } D. { X ∣ X = 13 } \{x \mid X = 13\} { X ∣ X = 13 }

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Introduction

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When dealing 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 will 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 is valid. 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 Composite Function $(f \circ g)(x)$

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To find the domain of the composite function $(f \circ g)(x)$, we need to first find the composite function itself. The composite function is defined as $(f \circ g)(x) = f(g(x))$. In this case, we have:

$(f \circ g)(x) = f(g(x)) = f\left(\frac{1}{x-13}\right) = \frac{1}{x-13} + 7$

Finding the Domain of the Composite Function

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To find the domain of the composite function, we need to identify any restrictions on the input values of $x$. In this case, we have two restrictions:

1. The denominator of the function cannot be zero. In other words, $x-13 \neq 0$.
2. The function is undefined when the denominator is zero, which occurs when $x = 13$.

Solving the Inequality

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To find the domain of the composite function, we need to solve the inequality $x-13 \neq 0$. We can do this by adding $13$ to both sides of the inequality:

$x-13+13 \neq 0+13$

This simplifies to:

$x \neq 13$

Conclusion

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In conclusion, the domain of the composite function $(f \circ g)(x)$ is all real numbers except $13$. This can be written as:

$\{x \mid x \neq 13\}$

Answer Key

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The correct answer is:

C. $\{x \mid x \neq -13\}$

Note: This answer is incorrect. The correct answer is $\{x \mid x \neq 13\}$.

Final Answer

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The final answer is $\boxed{\{x \mid x \neq 13\}}$.

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Introduction

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In our previous article, we explored the concept of the domain of a composite function and found the domain of the composite function $(f \circ g)(x)$, where $f(x) = x + 7$ and $g(x) = \frac{1}{x-13}$. In this article, we will provide a Q&A guide to help you understand the concept of the domain of a composite function.

Q&A

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Q1: What is the domain of a composite function?

A1: The domain of a composite function is the set of all possible input values for which the function is defined.

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

A2: To find the domain of a composite function, you need to identify any restrictions on the input values of $x$. This can be done by analyzing the function and identifying any values of $x$ that would make the function undefined.

Q3: What are some common restrictions on the input values of $x$?

A3: Some common restrictions on the input values of $x$ include:

• The denominator of the function cannot be zero.
• The function is undefined when the denominator is zero.
• The function is undefined when the input value is outside the range of the function.

Q4: How do I solve an inequality to find the domain of a composite function?

A4: To solve an inequality to find the domain of a composite function, you can use the following steps:

1. Identify the inequality that needs to be solved.
2. Add or subtract the same value to both sides of the inequality to isolate the variable.
3. Simplify the inequality to find the solution.

Q5: What is the domain of the composite function $(f \circ g)(x)$, where $f(x) = x + 7$ and $g(x) = \frac{1}{x-13}$?

A5: 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 $13$.

Q6: How do I write the domain of a composite function in interval notation?

A6: To write the domain of a composite function in interval notation, you can use the following steps:

1. Identify the domain of the composite function.
2. Write the domain in interval notation using the following format: $[a, b)$, $(a, b]$, $[a, b]$, or $(a, b)$.

Q7: What is the difference between the domain and the range of a function?

A7: The domain of a function is the set of all possible input values for which the function is defined, while the range of a function is the set of all possible output values of the function.

Q8: How do I find the range of a composite function?

A8: To find the range of a composite function, you need to analyze the function and identify the possible output values. This can be done by analyzing the function and identifying the minimum and maximum values of the function.

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 identify any restrictions on the input values of $x$ and solve any inequalities that arise. The domain of a composite function can be written in interval notation using the following format: $[a, b)$, $(a, b]$, $[a, b]$, or $(a, b)$.

Final Answer

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The final answer is $\boxed{\{x \mid x \neq 13\}}$.