Given The Functions \[$ F(x) = 5 - X \$\] And \[$ G(x) = 3x + 7 \$\],(a) Simplify \[$ F(2x) + G(x-1) \$\].

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Introduction

In mathematics, composite functions are a fundamental concept that plays a crucial role in various mathematical operations. Given two functions, f(x) and g(x), we can create a new function by combining them in different ways. In this article, we will explore how to simplify composite functions using the given functions f(x) = 5 - x and g(x) = 3x + 7.

Understanding Composite Functions

A composite function is a function that is derived from two or more functions. It is created by substituting one function into another function. For example, if we have two functions f(x) and g(x), we can create a new function by substituting g(x) into f(x). This new function is called a composite function.

Simplifying Composite Functions

To simplify a composite function, we need to follow a step-by-step approach. Let's consider the given functions f(x) = 5 - x and g(x) = 3x + 7. We are asked to simplify the expression f(2x) + g(x-1).

Step 1: Substitute 2x into f(x)

First, we need to substitute 2x into f(x). This means we replace x with 2x in the function f(x) = 5 - x.

f(2x) = 5 - 2x

Step 2: Substitute x-1 into g(x)

Next, we need to substitute x-1 into g(x). This means we replace x with x-1 in the function g(x) = 3x + 7.

g(x-1) = 3(x-1) + 7

Step 3: Simplify g(x-1)

Now, we need to simplify the expression g(x-1) = 3(x-1) + 7.

g(x-1) = 3x - 3 + 7 g(x-1) = 3x + 4

Step 4: Add f(2x) and g(x-1)

Finally, we need to add f(2x) and g(x-1).

f(2x) + g(x-1) = (5 - 2x) + (3x + 4)

Step 5: Simplify the Expression

Now, we need to simplify the expression (5 - 2x) + (3x + 4).

(5 - 2x) + (3x + 4) = 5 - 2x + 3x + 4 (5 - 2x) + (3x + 4) = 5 + x + 4 (5 - 2x) + (3x + 4) = 9 + x

Conclusion

In this article, we have learned how to simplify composite functions using the given functions f(x) = 5 - x and g(x) = 3x + 7. We have followed a step-by-step approach to simplify the expression f(2x) + g(x-1). By substituting 2x into f(x) and x-1 into g(x), we have simplified the expression to 9 + x. This demonstrates the importance of understanding composite functions and how to simplify them using algebraic manipulations.

Example Problems

  1. Simplify the expression f(3x) + g(x+2) using the functions f(x) = 2x + 1 and g(x) = x^2 - 3.
  2. Simplify the expression f(x/2) + g(2x) using the functions f(x) = x^2 + 1 and g(x) = 2x - 1.

Tips and Tricks

  • When simplifying composite functions, always follow the order of operations (PEMDAS).
  • Use algebraic manipulations to simplify the expression.
  • Substitute the functions into each other to create a new function.
  • Simplify the new function using algebraic manipulations.

Common Mistakes

  • Failing to follow the order of operations (PEMDAS).
  • Not simplifying the expression using algebraic manipulations.
  • Not substituting the functions into each other to create a new function.

Real-World Applications

Composite functions have numerous real-world applications in various fields such as physics, engineering, and economics. For example, in physics, composite functions are used to model complex systems and predict their behavior. In engineering, composite functions are used to design and optimize systems. In economics, composite functions are used to model economic systems and predict their behavior.

Conclusion

Introduction

In our previous article, we explored how to simplify composite functions using the given functions f(x) = 5 - x and g(x) = 3x + 7. We followed a step-by-step approach to simplify the expression f(2x) + g(x-1). In this article, we will answer some frequently asked questions about simplifying composite functions.

Q&A

Q: What is a composite function?

A: A composite function is a function that is derived from two or more functions. It is created by substituting one function into another function.

Q: How do I simplify a composite function?

A: To simplify a composite function, you need to follow a step-by-step approach. First, substitute the functions into each other to create a new function. Then, simplify the new function using algebraic manipulations.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying an expression. The acronym PEMDAS stands for:

  • Parentheses: Evaluate expressions inside parentheses first.
  • Exponents: Evaluate any exponential expressions next.
  • Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  • Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I know which function to substitute into the other?

A: When simplifying a composite function, you need to substitute the functions into each other in a way that creates a new function. This means that you need to substitute the input of one function into the other function.

Q: Can I simplify a composite function using a calculator?

A: While a calculator can be a useful tool for simplifying expressions, it is not always the best way to simplify a composite function. When simplifying a composite function, it is often better to use algebraic manipulations to simplify the expression.

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

A: Some common mistakes to avoid when simplifying composite functions include:

  • Failing to follow the order of operations (PEMDAS).
  • Not simplifying the expression using algebraic manipulations.
  • Not substituting the functions into each other to create a new function.

Q: How do I know if a composite function is simplified?

A: A composite function is simplified when it can be written in a simple form, such as a linear or quadratic function. To check if a composite function is simplified, you can try to simplify it using algebraic manipulations.

Q: Can I use composite functions to model real-world systems?

A: Yes, composite functions can be used to model real-world systems. For example, in physics, composite functions are used to model complex systems and predict their behavior. In engineering, composite functions are used to design and optimize systems.

Example Problems

  1. Simplify the expression f(3x) + g(x+2) using the functions f(x) = 2x + 1 and g(x) = x^2 - 3.
  2. Simplify the expression f(x/2) + g(2x) using the functions f(x) = x^2 + 1 and g(x) = 2x - 1.

Tips and Tricks

  • When simplifying composite functions, always follow the order of operations (PEMDAS).
  • Use algebraic manipulations to simplify the expression.
  • Substitute the functions into each other to create a new function.
  • Simplify the new function using algebraic manipulations.

Common Mistakes

  • Failing to follow the order of operations (PEMDAS).
  • Not simplifying the expression using algebraic manipulations.
  • Not substituting the functions into each other to create a new function.

Real-World Applications

Composite functions have numerous real-world applications in various fields such as physics, engineering, and economics. For example, in physics, composite functions are used to model complex systems and predict their behavior. In engineering, composite functions are used to design and optimize systems. In economics, composite functions are used to model economic systems and predict their behavior.

Conclusion

In conclusion, simplifying composite functions is an essential skill in mathematics that has numerous real-world applications. By following a step-by-step approach and using algebraic manipulations, we can simplify complex expressions and create new functions. This article has demonstrated the importance of understanding composite functions and how to simplify them using algebraic manipulations.