If M ( X ) = X + 5 X − 1 M(x)=\frac{x+5}{x-1} M ( X ) = X − 1 X + 5 ​ And N ( X ) = X − 3 N(x)=x-3 N ( X ) = X − 3 , Which Function Has The Same Domain As ( M ∘ N ) ( X (m \circ N)(x ( M ∘ N ) ( X ]?A. H ( X ) = X + 5 11 H(x)=\frac{x+5}{11} H ( X ) = 11 X + 5 ​ B. H ( X ) = 11 X − 1 H(x)=\frac{11}{x-1} H ( X ) = X − 1 11 ​ C. H ( X ) = 11 X − 4 H(x)=\frac{11}{x-4} H ( X ) = X − 4 11 ​ D. H ( X ) = 11 X − 3 H(x)=\frac{11}{x-3} H ( X ) = X − 3 11 ​

In mathematics, composite functions are a crucial concept in understanding the behavior of functions. When dealing with composite functions, it's essential to analyze their domains to ensure that they are well-defined. In this article, we will explore the domain analysis of composite functions, specifically focusing on the given functions $m(x)$ and $n(x)$.

Understanding Composite Functions

A composite function is a function that is derived from the composition of two or more functions. The composition of two functions $f(x)$ and $g(x)$ is denoted as $(f \circ g)(x)$ and is defined as $f(g(x))$. In other words, the output of the inner function $g(x)$ is used as the input for the outer function $f(x)$.

Given Functions

We are given two functions:

• $m(x) = \frac{x+5}{x-1}$
• $n(x) = x - 3$

We are asked to determine which function has the same domain as $(m \circ n)(x)$.

Domain of a Function

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 well-defined.

Domain of $m(x)$

The function $m(x) = \frac{x+5}{x-1}$ is defined as long as the denominator $x-1$ is not equal to zero. Therefore, the domain of $m(x)$ is all real numbers except $x=1$.

Domain of $n(x)$

The function $n(x) = x - 3$ is defined for all real numbers, as there are no restrictions on the input values.

Composite Function $(m \circ n)(x)$

To find the composite function $(m \circ n)(x)$, we substitute $n(x)$ into $m(x)$:

$(m \circ n)(x) = m(n(x)) = \frac{(x-3)+5}{(x-3)-1} = \frac{x+2}{x-4}$

Domain of $(m \circ n)(x)$

The function $(m \circ n)(x) = \frac{x+2}{x-4}$ is defined as long as the denominator $x-4$ is not equal to zero. Therefore, the domain of $(m \circ n)(x)$ is all real numbers except $x=4$.

Comparing Domains

We are asked to determine which function has the same domain as $(m \circ n)(x)$. To do this, we need to compare the domains of the given functions with the domain of $(m \circ n)(x)$.

• The domain of $h(x) = \frac{x+5}{11}$ is all real numbers, as there are no restrictions on the input values.
• The domain of $h(x) = \frac{11}{x-1}$ is all real numbers except $x=1$.
• The domain of $h(x) = \frac{11}{x-4}$ is all real numbers except $x=4$.
• The domain of $h(x) = \frac{11}{x-3}$ is all real numbers except $x=3$.

Conclusion

Based on the analysis, we can conclude that the function with the same domain as $(m \circ n)(x)$ is:

• $h(x) = \frac{11}{x-4}$

This is because the domain of $h(x) = \frac{11}{x-4}$ is all real numbers except $x=4$, which is the same as the domain of $(m \circ n)(x)$.

Final Answer

The final answer is:

• $h(x) = \frac{11}{x-4}$
  Q&A: Domain Analysis of Composite Functions =============================================

In the previous article, we explored the domain analysis of composite functions, specifically focusing on the given functions $m(x)$ and $n(x)$. In this article, we will answer some frequently asked questions related to domain analysis of composite functions.

Q: What is the domain of a composite function?

A: The domain of a composite 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 well-defined.

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

A: To find the domain of a composite function, you need to analyze the domains of the individual functions and determine the restrictions on the input values. You can do this by:

• Finding the domain of the inner function
• Finding the domain of the outer function
• Determining the restrictions on the input values

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

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.

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

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: What is the domain of a function with a square root?

A: The domain of a function with a square root is all real numbers except the values that make the expression inside the square root negative.

Q: How do I determine the domain of a composite function with a square root?

A: To determine the domain of a composite function with a square root, you need to analyze the domains of the individual functions and determine the restrictions on the input values. You can do this by:

• Finding the domain of the inner function
• Finding the domain of the outer function
• Determining the restrictions on the input values

Q: What is the importance of domain analysis in mathematics?

A: Domain analysis is crucial in mathematics as it helps us understand the behavior of functions and determine the restrictions on the input values. It's essential in various areas of mathematics, including algebra, calculus, and analysis.

Q: How do I apply domain analysis in real-world problems?

A: Domain analysis is applied in various real-world problems, including:

• Physics: To determine the domain of a function that represents a physical quantity, such as velocity or acceleration.
• Engineering: To determine the domain of a function that represents a physical quantity, such as stress or strain.
• Economics: To determine the domain of a function that represents a economic quantity, such as demand or supply.

Conclusion

In conclusion, domain analysis is a crucial concept in mathematics that helps us understand the behavior of functions and determine the restrictions on the input values. By analyzing the domains of individual functions and determining the restrictions on the input values, we can determine the domain of a composite function. We hope this article has provided you with a better understanding of domain analysis and its importance in mathematics.

Final Answer

The final answer is:

• 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 analyze the domains of the individual functions and determine the restrictions on the input values.
• The domain of a function with a square root is all real numbers except the values that make the expression inside the square root negative.
• Domain analysis is crucial in mathematics as it helps us understand the behavior of functions and determine the restrictions on the input values.