1. Let $f(x)=2x+1$ And $g(x)=3x^2$. Which Of The Following Represents The Composition Function $g(f(x)$\]?A. $6x+3$ B. $6x^2+1$ C. $12x^2+3$ D. $12x^2+12x+3$

Introduction

In mathematics, a composition of functions is a way of combining two or more functions to create a new function. This is a fundamental concept in algebra and calculus, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore the concept of composition of functions, and we will use the given functions $f(x)=2x+1$ and $g(x)=3x^2$ to illustrate the process.

What is a Composition of Functions?

A composition of functions is a way of combining two or more functions to create a new function. This is done by plugging one function into another function. In other words, we take the output of one function and use it as the input for another function. The resulting function is called the composition function.

Notation

The composition of functions is denoted by the symbol $\circ$. For example, if we have two functions $f(x)$ and $g(x)$, the composition function $g(f(x))$ is denoted by $g \circ f$. This notation indicates that we are plugging the function $f(x)$ into the function $g(x)$.

Step-by-Step Guide to Composition of Functions

To find the composition function $g(f(x))$, we need to follow these steps:

1. Plug the function $f(x)$ into the function $g(x)$: This means that we will replace the variable $x$ in the function $g(x)$ with the function $f(x)$.
2. Simplify the resulting expression: After plugging the function $f(x)$ into the function $g(x)$, we need to simplify the resulting expression.

Example: Composition of Functions

Let's use the given functions $f(x)=2x+1$ and $g(x)=3x^2$ to illustrate the process of composition of functions.

Step 1: Plug the function $f(x)$ into the function $g(x)$

We will replace the variable $x$ in the function $g(x)$ with the function $f(x)$.

$g(f(x)) = 3(f(x))^2$

Step 2: Simplify the resulting expression

Now, we will simplify the resulting expression by plugging in the function $f(x)=2x+1$.

$g(f(x)) = 3(2x+1)^2$

$g(f(x)) = 3(4x^2+4x+1)$

$g(f(x)) = 12x^2+12x+3$

Conclusion

In this article, we have explored the concept of composition of functions and used the given functions $f(x)=2x+1$ and $g(x)=3x^2$ to illustrate the process. We have shown that the composition function $g(f(x))$ is equal to $12x^2+12x+3$. This is a fundamental concept in algebra and calculus, and it has numerous applications in various fields.

Answer

The correct answer is D. $12x^2+12x+3$.

Discussion

This problem is a great example of how composition of functions can be used to create new functions. The concept of composition of functions is a fundamental concept in algebra and calculus, and it has numerous applications in various fields. In this problem, we have used the given functions $f(x)=2x+1$ and $g(x)=3x^2$ to illustrate the process of composition of functions. We have shown that the composition function $g(f(x))$ is equal to $12x^2+12x+3$. This is a great example of how composition of functions can be used to create new functions.

Related Topics

• Function Composition: This is a fundamental concept in algebra and calculus, and it has numerous applications in various fields.
• Algebra: This is a branch of mathematics that deals with the study of algebraic structures, such as groups, rings, and fields.
• Calculus: This is a branch of mathematics that deals with the study of rates of change and accumulation.
• Mathematical Functions: This is a branch of mathematics that deals with the study of functions, including their properties and behavior.

References

• "Algebra" by Michael Artin: This is a comprehensive textbook on algebra that covers the basics of algebra, including function composition.
• "Calculus" by Michael Spivak: This is a comprehensive textbook on calculus that covers the basics of calculus, including function composition.
• "Mathematical Functions" by Walter Rudin: This is a comprehensive textbook on mathematical functions that covers the basics of mathematical functions, including function composition.
  Composition of Functions: A Q&A Guide =====================================

Introduction

In our previous article, we explored the concept of composition of functions and used the given functions $f(x)=2x+1$ and $g(x)=3x^2$ to illustrate the process. In this article, we will answer some frequently asked questions about composition of functions.

Q: What is the composition of functions?

A: The composition of functions is a way of combining two or more functions to create a new function. This is done by plugging one function into another function.

Q: How do I find the composition function?

A: To find the composition function, you need to follow these steps:

1. Plug the function $f(x)$ into the function $g(x)$: This means that you will replace the variable $x$ in the function $g(x)$ with the function $f(x)$.
2. Simplify the resulting expression: After plugging the function $f(x)$ into the function $g(x)$, you need to simplify the resulting expression.

Q: What is the notation for composition of functions?

A: The notation for composition of functions is denoted by the symbol $\circ$. For example, if we have two functions $f(x)$ and $g(x)$, the composition function $g(f(x))$ is denoted by $g \circ f$.

Q: How do I know if a function is a composition of functions?

A: To determine if a function is a composition of functions, you need to check if it can be written as the composition of two or more functions. If it can be written in this form, then it is a composition of functions.

Q: What are some examples of composition of functions?

A: Here are some examples of composition of functions:

• $f(x) = 2x + 1$ and $g(x) = 3x^2$: The composition function $g(f(x))$ is equal to $12x^2 + 12x + 3$.
• $f(x) = x^2 + 1$ and $g(x) = 2x - 1$: The composition function $g(f(x))$ is equal to $2x^4 + 4x^2 + 1$.
• $f(x) = x^3 - 1$ and $g(x) = x + 1$: The composition function $g(f(x))$ is equal to $x^4 + 3x^3 - 2x^2 + 2x$.

Q: What are some real-world applications of composition of functions?

A: Composition of functions has numerous real-world applications in various fields, including:

• Physics: Composition of functions is used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Composition of functions is used to design and analyze complex systems, such as electrical circuits and mechanical systems.
• Economics: Composition of functions is used to model economic systems and make predictions about economic trends.

Q: How do I practice composition of functions?

A: To practice composition of functions, you can try the following:

• Work through examples: Try working through examples of composition of functions to get a feel for how it works.
• Practice with different functions: Try practicing with different functions to see how composition of functions works with different types of functions.
• Use online resources: There are many online resources available that can help you practice composition of functions, including video tutorials and practice problems.

Conclusion

In this article, we have answered some frequently asked questions about composition of functions. We have also provided some examples of composition of functions and discussed some real-world applications of this concept. By practicing composition of functions, you can develop a deeper understanding of this important mathematical concept.