If F ( X ) = X F(x) = X F ( X ) = X And G ( X ) = 2 G(x) = 2 G ( X ) = 2 , What Is ( F ∘ G ) ( X (f \circ G)(x ( F ∘ G ) ( X ]?A. ( F ∘ G ) ( X ) = 2 X (f \circ G)(x) = 2x ( F ∘ G ) ( X ) = 2 X B. ( F ∘ G ) ( X ) = 2 (f \circ G)(x) = 2 ( F ∘ G ) ( X ) = 2 C. ( F ∘ G ) ( X ) = X + 2 (f \circ G)(x) = X + 2 ( F ∘ G ) ( X ) = X + 2 D. ( F ∘ G ) ( X ) = X 2 (f \circ G)(x) = \frac{x}{2} ( F ∘ G ) ( X ) = 2 X ​

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Introduction to Function Composition

In mathematics, function composition is a fundamental concept that allows us to combine two or more functions to create a new function. This new function takes the output of one function as the input for another function. The composition of functions is denoted by the symbol $\circ$, and it is read as "function $f$ composed with function $g$." In this article, we will explore the concept of function composition and use it to solve a specific problem involving the functions $f(x) = x$ and $g(x) = 2$.

Understanding the Composition of Functions

To understand the composition of functions, let's consider a simple example. Suppose we have two functions, $f(x) = x^2$ and $g(x) = x + 1$. We can compose these functions by plugging the output of $g(x)$ into the input of $f(x)$. This gives us the new function $(f \circ g)(x) = f(g(x)) = (x + 1)^2$. In this example, we can see that the composition of functions involves replacing the input variable $x$ in the function $f(x)$ with the output of the function $g(x)$.

Composition of Functions: A Step-by-Step Approach

Now that we have a basic understanding of function composition, let's apply this concept to the problem at hand. We are given two functions, $f(x) = x$ and $g(x) = 2$. We want to find the composition of these functions, denoted by $(f \circ g)(x)$. To do this, we need to plug the output of $g(x)$ into the input of $f(x)$. Since $g(x) = 2$, we can substitute this value into the function $f(x)$ to get $(f \circ g)(x) = f(g(x)) = f(2)$.

Solving for $(f \circ g)(x)$

Now that we have the expression $(f \circ g)(x) = f(2)$, we can simplify this expression by plugging in the value of $f(x) = x$. This gives us $(f \circ g)(x) = 2$. Therefore, the composition of the functions $f(x) = x$ and $g(x) = 2$ is $(f \circ g)(x) = 2$.

Conclusion

In this article, we explored the concept of function composition and used it to solve a specific problem involving the functions $f(x) = x$ and $g(x) = 2$. We saw that the composition of functions involves replacing the input variable $x$ in the function $f(x)$ with the output of the function $g(x)$. By applying this concept, we were able to find the composition of the given functions and simplify the expression to get the final answer.

Answer

The final answer to the problem is:

$(f \circ g)(x) = 2$

This answer is consistent with option B, which states that $(f \circ g)(x) = 2$.

Additional Examples

To further illustrate the concept of function composition, let's consider a few additional examples.

Example 1

Suppose we have two functions, $f(x) = x^2$ and $g(x) = x + 1$. We can compose these functions by plugging the output of $g(x)$ into the input of $f(x)$. This gives us the new function $(f \circ g)(x) = f(g(x)) = (x + 1)^2$.

Example 2

Suppose we have two functions, $f(x) = \sin x$ and $g(x) = 2x$. We can compose these functions by plugging the output of $g(x)$ into the input of $f(x)$. This gives us the new function $(f \circ g)(x) = f(g(x)) = \sin (2x)$.

Example 3

Suppose we have two functions, $f(x) = x^3$ and $g(x) = x - 1$. We can compose these functions by plugging the output of $g(x)$ into the input of $f(x)$. This gives us the new function $(f \circ g)(x) = f(g(x)) = (x - 1)^3$.

These examples illustrate the concept of function composition and demonstrate how it can be used to create new functions from existing ones.

Final Thoughts

In conclusion, function composition is a powerful tool that allows us to create new functions from existing ones. By understanding the concept of function composition, we can solve a wide range of problems in mathematics and other fields. In this article, we explored the concept of function composition and used it to solve a specific problem involving the functions $f(x) = x$ and $g(x) = 2$. We saw that the composition of functions involves replacing the input variable $x$ in the function $f(x)$ with the output of the function $g(x)$. By applying this concept, we were able to find the composition of the given functions and simplify the expression to get the final answer.

Introduction

In our previous article, we explored the concept of function composition and used it to solve a specific problem involving the functions $f(x) = x$ and $g(x) = 2$. In this article, we will continue to delve deeper into the world of function composition and answer some of the most frequently asked questions about this topic.

Q&A: Function Composition

Q: What is function composition?

A: Function composition is a way of combining two or more functions to create a new function. This new function takes the output of one function as the input for another function.

Q: How do I find the composition of two functions?

A: To find the composition of two functions, you need to plug the output of one function into the input of the other function. This can be represented mathematically as $(f \circ g)(x) = f(g(x))$.

Q: What is the difference between function composition and function evaluation?

A: Function composition involves combining two or more functions to create a new function, whereas function evaluation involves finding the value of a function at a specific point.

Q: Can I compose more than two functions?

A: Yes, you can compose more than two functions. For example, if you have three functions $f(x)$, $g(x)$, and $h(x)$, you can compose them as $(f \circ g \circ h)(x) = f(g(h(x)))$.

Q: How do I simplify a composition of functions?

A: To simplify a composition of functions, you need to apply the rules of algebra and simplify the expression. For example, if you have the composition $(f \circ g)(x) = f(g(x)) = (x + 1)^2$, you can simplify it by expanding the square to get $(x + 1)^2 = x^2 + 2x + 1$.

Q: Can I use function composition to solve real-world problems?

A: Yes, function composition can be used to solve a wide range of real-world problems. For example, in physics, you can use function composition to model the motion of an object under the influence of gravity. In economics, you can use function composition to model the relationship between supply and demand.

Q: What are some common applications of function composition?

A: Some common applications of function composition include:

• Modeling real-world phenomena, such as population growth or economic systems
• Solving optimization problems, such as finding the maximum or minimum of a function
• Creating new functions from existing ones, such as combining two or more functions to create a new function

Q: How do I know when to use function composition?

A: You should use function composition when you need to combine two or more functions to create a new function. This can be useful when you need to model complex real-world phenomena or solve optimization problems.

Conclusion

In this article, we answered some of the most frequently asked questions about function composition. We explored the concept of function composition and its applications, and provided examples of how to simplify a composition of functions. By understanding function composition, you can solve a wide range of problems in mathematics and other fields.

Final Thoughts

Function composition is a powerful tool that allows us to create new functions from existing ones. By understanding the concept of function composition, we can solve a wide range of problems in mathematics and other fields. In this article, we explored the concept of function composition and its applications, and provided examples of how to simplify a composition of functions. We hope that this article has been helpful in your understanding of function composition.

Additional Resources

If you want to learn more about function composition, we recommend checking out the following resources:

• Khan Academy: Function Composition
• MIT OpenCourseWare: Function Composition
• Wolfram MathWorld: Function Composition

These resources provide a comprehensive introduction to function composition and its applications.