Select The Correct Answer.Which Function Defines \[$(g \cdot F)(x)\$\]?$\[ \begin{array}{l} f(x) = \log (5x) \\ g(x) = 5x + 4 \end{array} \\]A. \[$(g \cdot F)(x) = 5x \log (5x) + 4\$\]B. \[$(g \cdot F)(x) = 5x - 4 - \log

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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 determine the correct answer to the given problem.

What is Composition of Functions?

The composition of functions is a way of combining two or more functions to create a new function. Given two functions f(x) and g(x), the composition of f and g is denoted by (g ∘ f)(x) or (f ∘ g)(x). The composition of functions is defined as:

(g ∘ f)(x) = g(f(x))

In other words, we first apply the function f to the input x, and then apply the function g to the result.

Example: Composition of Functions

Let's consider two functions:

f(x) = log(5x) g(x) = 5x + 4

We want to find the composition of f and g, denoted by (g ∘ f)(x). To do this, we first apply the function f to the input x, and then apply the function g to the result.

First, we apply the function f to the input x:

f(x) = log(5x)

Next, we apply the function g to the result:

g(f(x)) = g(log(5x)) = 5(log(5x)) + 4

Now, we can simplify the expression:

g(f(x)) = 5(log(5x)) + 4 = 5(log(5x)) + 4

Answer:

Based on our calculation, we can see that the correct answer is:

A. (g ∘ f)(x) = 5(log(5x)) + 4

Why is this the Correct Answer?

The correct answer is (g ∘ f)(x) = 5(log(5x)) + 4 because we first applied the function f to the input x, and then applied the function g to the result. This is the definition of composition of functions.

What is the Composition of Functions Used For?

The composition of functions is used in various areas of mathematics, including algebra, calculus, and analysis. It is a powerful tool for solving problems and modeling real-world phenomena.

Example: Real-World Application

One real-world application of composition of functions is in physics. In physics, we often use functions to model the motion of objects. For example, we might use a function to model the position of an object as a function of time. We can then use the composition of functions to model the velocity of the object as a function of time.

Conclusion

In conclusion, the composition of functions is a fundamental concept in mathematics that allows us to combine two or more functions to create a new function. We have seen how to calculate the composition of functions and have determined the correct answer to the given problem. The composition of functions is a powerful tool for solving problems and modeling real-world phenomena.

References

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

Further Reading

If you want to learn more about composition of functions, I recommend checking out the following resources:

  • Khan Academy: Composition of Functions
  • Math Open Reference: Composition of Functions
  • Wolfram MathWorld: Composition of Functions

Final Answer

The final answer is:

Introduction

In our previous article, we explored the concept of composition of functions and determined the correct answer to the given problem. In this article, we will answer some frequently asked questions about composition of functions.

Q: What is the difference between (g ∘ f)(x) and (f ∘ g)(x)?

A: The difference between (g ∘ f)(x) and (f ∘ g)(x) is the order in which the functions are applied. (g ∘ f)(x) means that we first apply the function f to the input x, and then apply the function g to the result. On the other hand, (f ∘ g)(x) 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 know which function to apply first?

A: To determine which function to apply first, we need to look at the definition of the composition of functions. If the function g is applied first, then we need to look at the definition of g and see what it does to the input x. If the function f is applied first, then we need to look at the definition of f and see what it does to the input x.

Q: Can I apply more than two functions in a composition?

A: Yes, you can apply more than two functions in a composition. For example, if we have three functions f, g, and h, we can define the composition of f and g as (g ∘ f)(x), and then define the composition of g and h as (h ∘ g)(x). We can then define the composition of f and h as (h ∘ f)(x).

Q: How do I simplify a composition of functions?

A: To simplify a composition of functions, we need to apply the definition of the composition of functions and simplify the resulting expression. We can use algebraic manipulations, such as combining like terms and canceling out common factors, to simplify the expression.

Q: Can I use composition of functions to solve real-world problems?

A: Yes, you can use composition of functions to solve real-world problems. Composition of functions is a powerful tool for modeling complex systems and solving problems in fields such as physics, engineering, and economics.

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

A: Some common applications of composition of functions include:

  • Modeling population growth and decay
  • Modeling the motion of objects
  • Modeling the behavior of electrical circuits
  • Modeling the behavior of financial systems

Q: How do I know if a composition of functions is invertible?

A: To determine if a composition of functions is invertible, we need to check if the function is one-to-one and onto. If the function is one-to-one and onto, then it is invertible.

Q: Can I use composition of functions to solve optimization problems?

A: Yes, you can use composition of functions to solve optimization problems. Composition of functions is a powerful tool for finding the maximum or minimum of a function.

Conclusion

In conclusion, composition of functions is a powerful tool for solving problems and modeling real-world phenomena. We have answered some frequently asked questions about composition of functions and provided examples of how to use it to solve real-world problems.

References

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

Further Reading

If you want to learn more about composition of functions, I recommend checking out the following resources:

  • Khan Academy: Composition of Functions
  • Math Open Reference: Composition of Functions
  • Wolfram MathWorld: Composition of Functions

Final Answer

The final answer is:

  • (g ∘ f)(x) = g(f(x))
  • (f ∘ g)(x) = f(g(x))
  • Composition of functions can be used to solve real-world problems, such as modeling population growth and decay, modeling the motion of objects, and modeling the behavior of electrical circuits.