Select The Correct Answer.Consider These Functions:${ \begin{array}{l} f(x) = X + 1 \ g(x) = \frac{2}{x} \end{array} }$Which Polynomial Is Equivalent To { (f \circ G)(x)$}$?A. { \frac{2}{x+1}$}$B.

Introduction

In mathematics, the composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. This concept is crucial in various areas of mathematics, including algebra, calculus, and analysis. In this article, we will explore the composition of functions and how to find the equivalent polynomial for a given composition.

Understanding the Composition of Functions

The composition of functions is denoted by the symbol $\circ$. Given two functions $f(x)$ and $g(x)$, the composition of $f$ and $g$ is defined as $(f \circ g)(x) = f(g(x))$. This means that we first apply the function $g$ to the input $x$, and then apply the function $f$ to the result.

Example: Composition of Two Functions

Let's consider two functions:

$$\begin{array}{l} f(x) = x + 1 \\ g(x) = \frac{2}{x} \end{array}$$

We want to find the equivalent polynomial for the composition $(f \circ g)(x)$. To do this, we need to substitute the function $g(x)$ into the function $f(x)$.

Finding the Equivalent Polynomial

To find the equivalent polynomial, we substitute $g(x) = \frac{2}{x}$ into the function $f(x) = x + 1$. This gives us:

$$(f \circ g)(x) = f(g(x)) = f\left(\frac{2}{x}\right) = \frac{2}{x} + 1$$

Simplifying the Expression

We can simplify the expression by combining the terms:

$$(f \circ g)(x) = \frac{2}{x} + 1 = \frac{2 + x}{x}$$

Conclusion

In this article, we have explored the composition of functions and how to find the equivalent polynomial for a given composition. We have considered two functions $f(x) = x + 1$ and $g(x) = \frac{2}{x}$, and found the equivalent polynomial for the composition $(f \circ g)(x)$. The equivalent polynomial is $\frac{2 + x}{x}$.

Answer

The correct answer is:

$$\frac{2 + x}{x}$$

This is the equivalent polynomial for the composition $(f \circ g)(x)$.

Discussion

The composition of functions is a powerful tool in mathematics that allows us to create new functions from existing ones. In this article, we have seen how to find the equivalent polynomial for a given composition. This concept is crucial in various areas of mathematics, including algebra, calculus, and analysis.

Further Reading

For further reading on the composition of functions, we recommend the following resources:

• [1] "Composition of Functions" by Khan Academy
• [2] "Composition of Functions" by Math Is Fun
• [3] "Composition of Functions" by Wolfram MathWorld

These resources provide a comprehensive introduction to the composition of functions and how to find the equivalent polynomial for a given composition.

References

[1] Khan Academy. (n.d.). Composition of Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f4c7f:composition-of-functions

[2] Math Is Fun. (n.d.). Composition of Functions. Retrieved from https://www.mathisfun.com/algebra/composition-of-functions.html

[3] Wolfram MathWorld. (n.d.). Composition of Functions. Retrieved from https://mathworld.wolfram.com/CompositionofFunctions.html

Introduction

In our previous article, we explored the composition of functions and how to find the equivalent polynomial for a given composition. In this article, we will answer some frequently asked questions about the 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. It is denoted by the symbol $\circ$ and is defined as $(f \circ g)(x) = f(g(x))$. This means that we first apply the function $g$ to the input $x$, and then apply the function $f$ to the result.

Q: How do I find the equivalent polynomial for a given composition?

A: To find the equivalent polynomial for a given composition, you need to substitute the function $g(x)$ into the function $f(x)$. This will give you a new function that is the composition of the two original functions.

Q: What is the difference between the composition of functions and the product of functions?

A: The composition of functions and the product of functions are two different operations. The composition of functions is denoted by the symbol $\circ$ and is defined as $(f \circ g)(x) = f(g(x))$. The product of functions is denoted by the symbol $\cdot$ and is defined as $(f \cdot g)(x) = f(x) \cdot g(x)$.

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 evaluate the composition of functions?

A: To evaluate the composition of functions, you need to follow the order of operations. First, you need to evaluate the innermost function, and then work your way outwards.

Q: Can I use the composition of functions to solve equations?

A: Yes, you can use the composition of functions to solve equations. By composing two functions, you can create a new function that can be used to solve equations.

Q: What are some common applications of the composition of functions?

A: The composition of functions has many applications in mathematics and science. Some common applications include:

• Modeling population growth
• Analyzing financial data
• Solving differential equations
• Creating computer algorithms

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 look for the symbol $\circ$ in the function definition. If the function definition includes the symbol $\circ$, then it is a composition of functions.

Q: Can I use the composition of functions to create new functions?

A: Yes, you can use the composition of functions to create new functions. By composing two functions, you can create a new function that has different properties than the original functions.

Q: How do I use the composition of functions to solve problems?

A: To use the composition of functions to solve problems, you need to follow these steps:

1. Identify the functions that you want to compose.
2. Determine the order of operations.
3. Evaluate the innermost function.
4. Work your way outwards, evaluating each function in turn.
5. Use the final result to solve the problem.

Conclusion

In this article, we have answered some frequently asked questions about the composition of functions. We have covered topics such as how to find the equivalent polynomial for a given composition, the difference between the composition of functions and the product of functions, and how to use the composition of functions to solve problems.

Further Reading

For further reading on the composition of functions, we recommend the following resources:

• [1] "Composition of Functions" by Khan Academy
• [2] "Composition of Functions" by Math Is Fun
• [3] "Composition of Functions" by Wolfram MathWorld

These resources provide a comprehensive introduction to the composition of functions and how to use it to solve problems.

References

[1] Khan Academy. (n.d.). Composition of Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f4c7f:composition-of-functions

[2] Math Is Fun. (n.d.). Composition of Functions. Retrieved from https://www.mathisfun.com/algebra/composition-of-functions.html

[3] Wolfram MathWorld. (n.d.). Composition of Functions. Retrieved from https://mathworld.wolfram.com/CompositionofFunctions.html