Given: ${ F(x) = X - 7 }$ ${ H(x) = 2x + 3 }$Write The Rule For ${ H(f(x)) }$.A. ${ H(f(x)) = 2x - 11 }$ B. ${ H(f(x)) = 2x - 7 }$ C. ${ H(f(x)) = 2x - 4 }$ D. $[ H(f(x)) = 3x -

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Introduction

In mathematics, functions are used to describe relationships between variables. When we have two functions, we can combine them to create a new function. This process is called composition of functions. In this article, we will explore how to write the rule for the composition of two functions, given the rules for the individual functions.

What is Composition of Functions?

Composition of functions 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. In other words, we use the output of one function as the input for another function.

Given Functions

We are given two functions:

  • f(x) = x - 7
  • h(x) = 2x + 3

Writing the Rule for h(f(x))

To write the rule for h(f(x)), we need to substitute the output of f(x) into the input of h(x).

Step 1: Find the Output of f(x)

The output of f(x) is x - 7.

Step 2: Substitute the Output of f(x) into h(x)

Now, we substitute x - 7 into the input of h(x), which is 2x + 3.

h(f(x)) = 2(x - 7) + 3

Step 3: Simplify the Expression

To simplify the expression, we need to distribute the 2 to both terms inside the parentheses.

h(f(x)) = 2x - 14 + 3

h(f(x)) = 2x - 11

Conclusion

In conclusion, the rule for h(f(x)) is h(f(x)) = 2x - 11.

Discussion

Now, let's discuss the options given in the problem.

  • Option A: h(f(x)) = 2x - 11. This is the correct answer.
  • Option B: h(f(x)) = 2x - 7. This is incorrect because the output of f(x) is x - 7, not x.
  • Option C: h(f(x)) = 2x - 4. This is incorrect because the output of f(x) is x - 7, not x - 4.
  • Option D: h(f(x)) = 3x - 11. This is incorrect because the output of f(x) is x - 7, not 3x.

Final Answer

Introduction

In our previous article, we explored how to write the rule for the composition of two functions, given the rules for the individual functions. In this article, we will answer some frequently asked questions about composition of functions.

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

A: Function composition is a way of combining two or more functions to create a new function. Function evaluation, on the other hand, is the process of finding the output of a function for a given input.

Q: How do I know which function to evaluate first in a composition of functions?

A: When evaluating a composition of functions, you should evaluate the function on the inside first, and then evaluate the function on the outside with the output of the inside 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), g(x), and h(x), you can compose them as h(g(f(x))).

Q: How do I simplify a composition of functions?

A: To simplify a composition of functions, you can use the following steps:

  1. Evaluate the function on the inside first.
  2. Substitute the output of the inside function into the function on the outside.
  3. Simplify the resulting expression.

Q: What is the order of operations for composition of functions?

A: The order of operations for composition of functions is:

  1. Evaluate the function on the inside first.
  2. Substitute the output of the inside function into the function on the outside.
  3. Simplify the resulting expression.

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

A: Yes, you can use composition of functions to solve equations. For example, if you have an equation f(x) = g(x), you can use composition of functions to find the value of x.

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

A: To use composition of functions to solve equations, you can follow these steps:

  1. Evaluate the function on the inside first.
  2. Substitute the output of the inside function into the function on the outside.
  3. Simplify the resulting expression.
  4. Set the resulting expression equal to the original equation.
  5. Solve for x.

Q: What are some common mistakes to avoid when working with composition of functions?

A: Some common mistakes to avoid when working with composition of functions include:

  • Evaluating the function on the outside first instead of the inside function.
  • Failing to simplify the resulting expression.
  • Not following the order of operations.

Conclusion

In conclusion, composition of functions is a powerful tool for solving equations and simplifying expressions. By following the steps outlined in this article, you can master composition of functions and use it to solve a wide range of problems.

Final Tips

  • Practice, practice, practice! The more you practice composition of functions, the more comfortable you will become with it.
  • Use composition of functions to solve equations and simplify expressions.
  • Avoid common mistakes such as evaluating the function on the outside first instead of the inside function.

Common Composition of Functions Formulas

  • h(f(x)) = h(x - 7)
  • f(g(x)) = f(2x + 3)
  • g(h(x)) = g(2x + 3)

Composition of Functions Examples

  • h(f(x)) = 2(x - 7) + 3
  • f(g(x)) = x - 7(2x + 3)
  • g(h(x)) = 2(2x + 3) + 3