Two Functions Are Defined As Follows: $ F X \rightarrow 2x - 1 }$ ${ G: X \rightarrow 4x - 2 }$where { X $ $ Is A Real Number.a) Find { F(x+1) - 9 \leq 0 $}$ And Determine The Value Of { X $}$.

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Introduction

In mathematics, composite functions are a fundamental concept that allows us to combine two or more functions to create a new function. In this article, we will explore how to solve inequalities involving composite functions. We will define two functions, f(x) and g(x), and then use these functions to solve an inequality.

Defining the Functions

The two functions, f(x) and g(x), are defined as follows:

  • f(x) = 2x - 1
  • g(x) = 4x - 2

These functions take a real number x as input and produce a real number as output.

Solving the Inequality

We are given the inequality f(x+1) - 9 ≤ 0. To solve this inequality, we need to first find the value of f(x+1) and then set it less than or equal to 9.

Step 1: Find the Value of f(x+1)

To find the value of f(x+1), we need to substitute (x+1) into the function f(x).

f(x+1) = 2(x+1) - 1

Using the distributive property, we can simplify this expression:

f(x+1) = 2x + 2 - 1

Combine like terms:

f(x+1) = 2x + 1

Step 2: Set f(x+1) Less Than or Equal to 9

Now that we have found the value of f(x+1), we can set it less than or equal to 9:

2x + 1 ≤ 9

Subtract 1 from both sides:

2x ≤ 8

Divide both sides by 2:

x ≤ 4

Conclusion

In this article, we defined two functions, f(x) and g(x), and then used these functions to solve an inequality. We found the value of f(x+1) and then set it less than or equal to 9. By solving the inequality, we determined that the value of x is less than or equal to 4.

Example Use Cases

  1. Finance: In finance, composite functions can be used to model the behavior of financial instruments, such as stocks and bonds. For example, a financial analyst might use a composite function to model the relationship between the price of a stock and the interest rate.
  2. Science: In science, composite functions can be used to model the behavior of physical systems, such as the motion of a particle or the flow of a fluid. For example, a physicist might use a composite function to model the relationship between the position and velocity of a particle.
  3. Engineering: In engineering, composite functions can be used to model the behavior of complex systems, such as the behavior of a mechanical system or the flow of a fluid through a pipe. For example, an engineer might use a composite function to model the relationship between the pressure and flow rate of a fluid.

Tips and Tricks

  1. Use the distributive property: When simplifying expressions involving composite functions, use the distributive property to expand the expression.
  2. Combine like terms: When simplifying expressions involving composite functions, combine like terms to simplify the expression.
  3. Check your work: When solving inequalities involving composite functions, check your work by plugging in values to verify that the solution is correct.

Common Mistakes

  1. Forgetting to distribute: When simplifying expressions involving composite functions, forgetting to distribute the function can lead to incorrect solutions.
  2. Not combining like terms: When simplifying expressions involving composite functions, not combining like terms can lead to incorrect solutions.
  3. Not checking work: When solving inequalities involving composite functions, not checking work can lead to incorrect solutions.

Conclusion

Q: What is a composite function?

A: A composite function is a function that is formed by combining two or more functions. In the case of the inequality we solved earlier, the composite function was f(x+1), which is formed by substituting (x+1) into the function f(x).

Q: How do I know which function to use when solving an inequality?

A: When solving an inequality involving composite functions, you need to identify the outer function and the inner function. The outer function is the function that is being applied to the inner function. In the case of the inequality we solved earlier, the outer function was f(x) and the inner function was (x+1).

Q: What is the distributive property, and how do I use it when simplifying expressions involving composite functions?

A: The distributive property is a mathematical property that allows you to distribute a function to each term inside the parentheses. When simplifying expressions involving composite functions, you can use the distributive property to expand the expression. For example, if you have the expression f(x+1), you can use the distributive property to expand it as follows:

f(x+1) = f(x) + f(1)

Q: How do I combine like terms when simplifying expressions involving composite functions?

A: When simplifying expressions involving composite functions, you can combine like terms to simplify the expression. Like terms are terms that have the same variable and exponent. For example, if you have the expression 2x + 2x, you can combine the like terms as follows:

2x + 2x = 4x

Q: What is the difference between a linear function and a composite function?

A: A linear function is a function that can be written in the form f(x) = ax + b, where a and b are constants. A composite function, on the other hand, is a function that is formed by combining two or more functions. While linear functions are simple and easy to work with, composite functions can be more complex and require more advanced mathematical techniques to solve.

Q: Can I use composite functions to solve inequalities involving multiple variables?

A: Yes, you can use composite functions to solve inequalities involving multiple variables. However, you need to be careful when simplifying the expression and combining like terms. It's also important to check your work by plugging in values to verify that the solution is correct.

Q: What are some common mistakes to avoid when solving inequalities involving composite functions?

A: Some common mistakes to avoid when solving inequalities involving composite functions include:

  • Forgetting to distribute the function
  • Not combining like terms
  • Not checking work
  • Using the wrong function or variable
  • Making algebraic errors

Q: How can I practice solving inequalities involving composite functions?

A: You can practice solving inequalities involving composite functions by working through examples and exercises in a textbook or online resource. You can also try solving inequalities involving composite functions on your own, using a calculator or computer program to check your work.

Q: What are some real-world applications of solving inequalities involving composite functions?

A: Solving inequalities involving composite functions has many real-world applications, including:

  • Finance: Solving inequalities involving composite functions can be used to model the behavior of financial instruments, such as stocks and bonds.
  • Science: Solving inequalities involving composite functions can be used to model the behavior of physical systems, such as the motion of a particle or the flow of a fluid.
  • Engineering: Solving inequalities involving composite functions can be used to model the behavior of complex systems, such as the behavior of a mechanical system or the flow of a fluid through a pipe.

Conclusion

In conclusion, solving inequalities involving composite functions requires a clear understanding of the functions and the ability to simplify expressions. By following the steps outlined in this article, you can solve inequalities involving composite functions and determine the value of x. Remember to use the distributive property, combine like terms, and check your work to ensure that your solution is correct.