If F ( X ) = X − 1 X F(x)=\frac{x-1}{x} F ( X ) = X X − 1 ​ , G ( X ) = X − 1 G(x)=x-1 G ( X ) = X − 1 , And H ( X ) = X + 1 H(x)=x+1 H ( X ) = X + 1 , What Is { (g \circ H \circ F)(x)$}$?A. X − 1 X \frac{x-1}{x} X X − 1 ​ B. X X − 1 \frac{x}{x-1} X − 1 X ​ C. − X -x − X D. X X X

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Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. Composition of functions is a process of combining two or more functions to create a new function. In this article, we will explore the composition of functions and use it to find the value of (g∘h∘f)(x)(g \circ h \circ f)(x).

What is Composition of Functions?

Composition of functions is a process of combining two or more functions to create a new function. It is denoted by the symbol ∘\circ. For example, if we have two functions f(x)f(x) and g(x)g(x), then the composition of ff and gg is denoted by (g∘f)(x)(g \circ f)(x).

How to Find the Composition of Functions?

To find the composition of functions, we need to follow these steps:

  1. Identify the functions: Identify the two functions that we want to compose.
  2. Replace the input variable: Replace the input variable of the second function with the output of the first function.
  3. Simplify the expression: Simplify the resulting expression to get the final function.

Step-by-Step Solution

Step 1: Identify the Functions

We are given three functions:

  • f(x)=x−1xf(x)=\frac{x-1}{x}
  • g(x)=x−1g(x)=x-1
  • h(x)=x+1h(x)=x+1

We need to find the composition of gg, hh, and ff, which is denoted by (g∘h∘f)(x)(g \circ h \circ f)(x).

Step 2: Replace the Input Variable

To find the composition of gg, hh, and ff, we need to replace the input variable of hh with the output of ff, and then replace the input variable of gg with the output of hh.

So, we have:

(g∘h∘f)(x)=g(h(f(x)))(g \circ h \circ f)(x) = g(h(f(x)))

Step 3: Simplify the Expression

Now, we need to simplify the expression by replacing the input variable of hh with the output of ff, and then replacing the input variable of gg with the output of hh.

We have:

h(f(x))=h(x−1x)h(f(x)) = h\left(\frac{x-1}{x}\right)

=x−1x+1= \frac{x-1}{x} + 1

=x−1+xx= \frac{x-1+x}{x}

=2x−1x= \frac{2x-1}{x}

Now, we have:

g(h(f(x)))=g(2x−1x)g(h(f(x))) = g\left(\frac{2x-1}{x}\right)

=2x−1x−1= \frac{2x-1}{x} - 1

=2x−1−xx= \frac{2x-1-x}{x}

=x−1x= \frac{x-1}{x}

Therefore, we have:

(g∘h∘f)(x)=x−1x(g \circ h \circ f)(x) = \frac{x-1}{x}

Conclusion

In this article, we explored the composition of functions and used it to find the value of (g∘h∘f)(x)(g \circ h \circ f)(x). We followed the steps of identifying the functions, replacing the input variable, and simplifying the expression to get the final function. The final answer is x−1x\frac{x-1}{x}.

Final Answer

The final answer is x−1x\boxed{\frac{x-1}{x}}.

Discussion

This problem is a great example of how composition of functions can be used to solve complex problems. By breaking down the problem into smaller steps and using the composition of functions, we were able to find the value of (g∘h∘f)(x)(g \circ h \circ f)(x).

Related Problems

If you want to practice more problems on composition of functions, here are some related problems:

  • Find the composition of f(x)=x2f(x)=x^2 and g(x)=x+1g(x)=x+1.
  • Find the composition of f(x)=1xf(x)=\frac{1}{x} and g(x)=x2g(x)=x^2.
  • Find the composition of f(x)=x−1f(x)=x-1 and g(x)=x+1g(x)=x+1.

References

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

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Introduction

In our previous article, we explored the composition of functions and used it to find the value of (g∘h∘f)(x)(g \circ h \circ f)(x). In this article, we will answer some frequently asked questions about composition of functions.

Q&A

Q: What is composition of functions?

A: Composition of functions is a process of combining two or more functions to create a new function. It is denoted by the symbol ∘\circ. For example, if we have two functions f(x)f(x) and g(x)g(x), then the composition of ff and gg is denoted by (g∘f)(x)(g \circ f)(x).

Q: How do I find the composition of functions?

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

  1. Identify the functions: Identify the two functions that you want to compose.
  2. Replace the input variable: Replace the input variable of the second function with the output of the first function.
  3. Simplify the expression: Simplify the resulting expression to get the final function.

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

A: Composition of functions and function notation are two different concepts. Function notation is used to represent a function as a mathematical expression, while composition of functions is used to combine two or more functions to create a new function.

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)f(x), g(x)g(x), and h(x)h(x), then you can find the composition of ff, gg, and hh as (h∘g∘f)(x)(h \circ g \circ f)(x).

Q: How do I know which function to compose first?

A: When composing functions, you need to follow the order of operations. The function that is inside the parentheses is evaluated first, and then the function that is outside the parentheses is evaluated.

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

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

Q: What are some common mistakes to avoid when composing functions?

A: Some common mistakes to avoid when composing functions include:

  • Not following the order of operations: Make sure to follow the order of operations when composing functions.
  • Not simplifying the expression: Make sure to simplify the resulting expression to get the final function.
  • Not checking for domain restrictions: Make sure to check for domain restrictions when composing functions.

Conclusion

In this article, we answered some frequently asked questions about composition of functions. We hope that this article has helped you to understand composition of functions better and to avoid common mistakes.

Final Answer

The final answer is x−1x\boxed{\frac{x-1}{x}}.

Discussion

This problem is a great example of how composition of functions can be used to solve complex problems. By breaking down the problem into smaller steps and using the composition of functions, we were able to find the value of (g∘h∘f)(x)(g \circ h \circ f)(x).

Related Problems

If you want to practice more problems on composition of functions, here are some related problems:

  • Find the composition of f(x)=x2f(x)=x^2 and g(x)=x+1g(x)=x+1.
  • Find the composition of f(x)=1xf(x)=\frac{1}{x} and g(x)=x2g(x)=x^2.
  • Find the composition of f(x)=x−1f(x)=x-1 and g(x)=x+1g(x)=x+1.

References

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