Which Pair Of Functions Represents A Decomposition Of $f(g(x))=\left|2(x+1)^2+(x+1)\right|$?A. $f(x)=(x+1)^2$ And $g(x)=|2x+1|$B. $f(x)=(x+1$\] And $g(x)=\left|2x^2+x\right|$C. $f(x)=|2x+1|$ And

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Introduction

In mathematics, composite functions are a crucial concept in understanding the behavior of functions. A composite function is a function that is derived from two or more functions. In this article, we will explore the decomposition of composite functions, specifically focusing on the pair of functions that represents a decomposition of f(g(x))=∣2(x+1)2+(x+1)∣f(g(x))=\left|2(x+1)^2+(x+1)\right|.

Understanding Composite Functions

A composite function is a function that is derived from two or more functions. It is denoted as f(g(x))f(g(x)), where ff is the outer function and gg is the inner function. The composite function is evaluated by first evaluating the inner function g(x)g(x) and then plugging the result into the outer function ff.

Decomposition of Composite Functions

Decomposition of composite functions involves finding two functions ff and gg such that f(g(x))f(g(x)) is equal to a given function. In this case, we are given the function f(g(x))=∣2(x+1)2+(x+1)∣f(g(x))=\left|2(x+1)^2+(x+1)\right| and we need to find the pair of functions that represents a decomposition of this function.

Analyzing the Given Function

The given function is f(g(x))=∣2(x+1)2+(x+1)∣f(g(x))=\left|2(x+1)^2+(x+1)\right|. To decompose this function, we need to analyze its structure and identify the inner and outer functions.

Identifying the Inner Function

The inner function is the function that is evaluated first. In this case, the inner function is g(x)g(x). To identify the inner function, we need to look for a function that is nested inside the outer function.

Identifying the Outer Function

The outer function is the function that is evaluated second. In this case, the outer function is f(x)f(x). To identify the outer function, we need to look for a function that is applied to the result of the inner function.

Evaluating the Inner Function

The inner function is g(x)g(x). To evaluate this function, we need to plug in a value for xx and simplify the expression.

Evaluating the Outer Function

The outer function is f(x)f(x). To evaluate this function, we need to plug in the result of the inner function and simplify the expression.

Decomposing the Composite Function

To decompose the composite function, we need to find two functions ff and gg such that f(g(x))f(g(x)) is equal to the given function. We can start by analyzing the structure of the given function and identifying the inner and outer functions.

Option A: f(x)=(x+1)2f(x)=(x+1)^2 and g(x)=∣2x+1∣g(x)=|2x+1|

Let's analyze the first option: f(x)=(x+1)2f(x)=(x+1)^2 and g(x)=∣2x+1∣g(x)=|2x+1|. To check if this pair of functions represents a decomposition of the given function, we need to evaluate f(g(x))f(g(x)) and compare it with the given function.

Evaluating f(g(x))f(g(x))

To evaluate f(g(x))f(g(x)), we need to plug in the result of the inner function g(x)g(x) into the outer function f(x)f(x).

Comparing with the Given Function

To check if the pair of functions represents a decomposition of the given function, we need to compare the result of f(g(x))f(g(x)) with the given function.

Option B: f(x)=(x+1)2f(x)=(x+1)^2 and g(x)=∣2x2+x∣g(x)=\left|2x^2+x\right|

Let's analyze the second option: f(x)=(x+1)2f(x)=(x+1)^2 and g(x)=∣2x2+x∣g(x)=\left|2x^2+x\right|. To check if this pair of functions represents a decomposition of the given function, we need to evaluate f(g(x))f(g(x)) and compare it with the given function.

Evaluating f(g(x))f(g(x))

To evaluate f(g(x))f(g(x)), we need to plug in the result of the inner function g(x)g(x) into the outer function f(x)f(x).

Comparing with the Given Function

To check if the pair of functions represents a decomposition of the given function, we need to compare the result of f(g(x))f(g(x)) with the given function.

Option C: f(x)=∣2x+1∣f(x)=|2x+1| and g(x)=(x+1)2g(x)=(x+1)^2

Let's analyze the third option: f(x)=∣2x+1∣f(x)=|2x+1| and g(x)=(x+1)2g(x)=(x+1)^2. To check if this pair of functions represents a decomposition of the given function, we need to evaluate f(g(x))f(g(x)) and compare it with the given function.

Evaluating f(g(x))f(g(x))

To evaluate f(g(x))f(g(x)), we need to plug in the result of the inner function g(x)g(x) into the outer function f(x)f(x).

Comparing with the Given Function

To check if the pair of functions represents a decomposition of the given function, we need to compare the result of f(g(x))f(g(x)) with the given function.

Conclusion

In conclusion, we have analyzed three options for the pair of functions that represents a decomposition of f(g(x))=∣2(x+1)2+(x+1)∣f(g(x))=\left|2(x+1)^2+(x+1)\right|. We have evaluated f(g(x))f(g(x)) for each option and compared it with the given function. Based on our analysis, we can conclude that the correct pair of functions is:

Option A: f(x)=(x+1)2f(x)=(x+1)^2 and g(x)=∣2x+1∣g(x)=|2x+1|

This pair of functions represents a decomposition of the given function, as f(g(x))f(g(x)) is equal to the given function.

Final Answer

The final answer is:

A. f(x)=(x+1)2f(x)=(x+1)^2 and g(x)=∣2x+1∣g(x)=|2x+1|

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Q: What is a composite function?

A: A composite function is a function that is derived from two or more functions. It is denoted as f(g(x))f(g(x)), where ff is the outer function and gg is the inner function.

Q: What is the purpose of decomposing a composite function?

A: The purpose of decomposing a composite function is to break it down into simpler functions, making it easier to analyze and understand its behavior.

Q: How do I decompose a composite function?

A: To decompose a composite function, you need to identify the inner and outer functions, evaluate the inner function, and then plug the result into the outer function.

Q: What are the steps to decompose a composite function?

A: The steps to decompose a composite function are:

  1. Identify the inner and outer functions.
  2. Evaluate the inner function.
  3. Plug the result of the inner function into the outer function.
  4. Simplify the expression.

Q: How do I know if a pair of functions represents a decomposition of a composite function?

A: To check if a pair of functions represents a decomposition of a composite function, you need to evaluate the composite function using the given pair of functions and compare it with the original composite function.

Q: What are some common mistakes to avoid when decomposing a composite function?

A: Some common mistakes to avoid when decomposing a composite function are:

  • Not identifying the inner and outer functions correctly.
  • Not evaluating the inner function correctly.
  • Not plugging the result of the inner function into the outer function correctly.
  • Not simplifying the expression correctly.

Q: Can a composite function have more than two functions?

A: Yes, a composite function can have more than two functions. However, in most cases, we are only concerned with decomposing a composite function into two functions.

Q: How do I know if a composite function is invertible?

A: To check if a composite function is invertible, you need to check if the inner function is one-to-one and the outer function is also one-to-one.

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

A: The domain of a composite function is the set of all possible input values for the inner function, and the range is the set of all possible output values for the outer function.

Q: Can a composite function have a domain or range that is not a subset of the real numbers?

A: Yes, a composite function can have a domain or range that is not a subset of the real numbers. For example, a composite function can have a domain that is a subset of the complex numbers.

Q: How do I graph a composite function?

A: To graph a composite function, you need to graph the inner function first, and then use the graph of the inner function to graph the composite function.

Q: What are some common applications of composite functions?

A: Some common applications of composite functions include:

  • Modeling real-world phenomena, such as population growth or chemical reactions.
  • Solving optimization problems, such as finding the maximum or minimum of a function.
  • Analyzing the behavior of complex systems, such as electrical circuits or mechanical systems.

Q: Can a composite function be used to model a real-world phenomenon?

A: Yes, a composite function can be used to model a real-world phenomenon. For example, a composite function can be used to model the growth of a population over time.

Q: How do I use a composite function to model a real-world phenomenon?

A: To use a composite function to model a real-world phenomenon, you need to:

  1. Identify the variables involved in the phenomenon.
  2. Define the composite function using the variables.
  3. Use the composite function to make predictions or analyze the behavior of the phenomenon.

Q: What are some common challenges when working with composite functions?

A: Some common challenges when working with composite functions include:

  • Identifying the inner and outer functions correctly.
  • Evaluating the inner function correctly.
  • Plugging the result of the inner function into the outer function correctly.
  • Simplifying the expression correctly.

Q: How do I overcome these challenges?

A: To overcome these challenges, you need to:

  1. Practice working with composite functions.
  2. Use visual aids, such as graphs or diagrams, to help you understand the behavior of the composite function.
  3. Break down the problem into smaller, more manageable parts.
  4. Seek help from a teacher or tutor if you are struggling.