For F ( X ) = 2 X F(x) = 2x F ( X ) = 2 X And G ( X ) = X + 8 G(x) = X + 8 G ( X ) = X + 8 , Find The Following Functions:a. ( F ∘ G ) ( X (f \circ G)(x ( F ∘ G ) ( X ]b. ( G ∘ F ) ( X (g \circ F)(x ( G ∘ F ) ( X ]c. ( F ∘ G ) ( 4 (f \circ G)(4 ( F ∘ G ) ( 4 ]d. ( G ∘ F ) ( 4 (g \circ F)(4 ( G ∘ F ) ( 4 ]

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 fields, including calculus, algebra, and analysis. In this article, we will explore the composition of functions, focusing on the given functions $f(x) = 2x$ and $g(x) = x + 8$. We will find the composite functions $(f \circ g)(x)$, $(g \circ f)(x)$, $(f \circ g)(4)$, and $(g \circ f)(4)$.

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 $(f \circ g)(x)$ and is defined as:

$$(f \circ g)(x) = f(g(x))$$

This means that we first apply the function $g$ to the input $x$, and then apply the function $f$ to the result.

Finding the Composite Functions

a. $(f \circ g)(x)$

To find the composite function $(f \circ g)(x)$, we need to substitute $g(x)$ into $f(x)$.

$$f(g(x)) = 2(g(x))$$

Since $g(x) = x + 8$, we can substitute this into the equation above:

$$f(g(x)) = 2(x + 8)$$

Expanding the equation, we get:

$$f(g(x)) = 2x + 16$$

Therefore, the composite function $(f \circ g)(x)$ is:

$$(f \circ g)(x) = 2x + 16$$

b. $(g \circ f)(x)$

To find the composite function $(g \circ f)(x)$, we need to substitute $f(x)$ into $g(x)$.

$$g(f(x)) = (f(x)) + 8$$

Since $f(x) = 2x$, we can substitute this into the equation above:

$$g(f(x)) = (2x) + 8$$

Simplifying the equation, we get:

$$g(f(x)) = 2x + 8$$

Therefore, the composite function $(g \circ f)(x)$ is:

$$(g \circ f)(x) = 2x + 8$$

c. $(f \circ g)(4)$

To find the value of $(f \circ g)(4)$, we need to substitute $x = 4$ into the composite function $(f \circ g)(x)$.

$$(f \circ g)(4) = 2(4) + 16$$

Evaluating the expression, we get:

$$(f \circ g)(4) = 8 + 16$$

Simplifying the equation, we get:

$$(f \circ g)(4) = 24$$

d. $(g \circ f)(4)$

To find the value of $(g \circ f)(4)$, we need to substitute $x = 4$ into the composite function $(g \circ f)(x)$.

$$(g \circ f)(4) = 2(4) + 8$$

Evaluating the expression, we get:

$$(g \circ f)(4) = 8 + 8$$

Simplifying the equation, we get:

$$(g \circ f)(4) = 16$$

Conclusion

In this article, we have explored the composition of functions, focusing on the given functions $f(x) = 2x$ and $g(x) = x + 8$. We have found the composite functions $(f \circ g)(x)$, $(g \circ f)(x)$, $(f \circ g)(4)$, and $(g \circ f)(4)$. The composition of functions is a powerful tool in mathematics, allowing us to combine functions to create new functions. Understanding the composition of functions is essential in various fields, including calculus, algebra, and analysis.

Key Takeaways

• The composition of functions is a way of combining two or more functions to create a new function.
• The composite function $(f \circ g)(x)$ is defined as $f(g(x))$.
• The composite function $(g \circ f)(x)$ is defined as $g(f(x))$.
• The value of $(f \circ g)(4)$ is 24.
• The value of $(g \circ f)(4)$ is 16.

Further Reading

For further reading on the composition of functions, we recommend the following resources:

• Wikipedia: Composition of Functions
• Khan Academy: Composition of Functions
• Mathway: Composition of Functions
  Composition of Functions: A Q&A Guide =====================================

Introduction

In our previous article, we explored the composition of functions, focusing on the given functions $f(x) = 2x$ and $g(x) = x + 8$. We found the composite functions $(f \circ g)(x)$, $(g \circ f)(x)$, $(f \circ g)(4)$, and $(g \circ f)(4)$. In this article, we will answer some frequently asked questions about the composition of functions.

Q&A

Q: What is the composition of functions?

A: 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 $(f \circ g)(x)$ and is defined as:

$$(f \circ g)(x) = f(g(x))$$

Q: How do I find the composite function $(f \circ g)(x)$?

A: To find the composite function $(f \circ g)(x)$, you need to substitute $g(x)$ into $f(x)$. For example, if $f(x) = 2x$ and $g(x) = x + 8$, then:

$$(f \circ g)(x) = f(g(x)) = 2(g(x)) = 2(x + 8) = 2x + 16$$

Q: How do I find the composite function $(g \circ f)(x)$?

A: To find the composite function $(g \circ f)(x)$, you need to substitute $f(x)$ into $g(x)$. For example, if $f(x) = 2x$ and $g(x) = x + 8$, then:

$$(g \circ f)(x) = g(f(x)) = (f(x)) + 8 = 2x + 8$$

Q: What is the difference between $(f \circ g)(x)$ and $(g \circ f)(x)$?

A: The composite functions $(f \circ g)(x)$ and $(g \circ f)(x)$ are different. The order of the functions matters. In general, $(f \circ g)(x) \neq (g \circ f)(x)$.

Q: How do I evaluate the composite function $(f \circ g)(x)$ at a specific value of $x$?

A: To evaluate the composite function $(f \circ g)(x)$ at a specific value of $x$, you need to substitute the value of $x$ into the composite function. For example, if $(f \circ g)(x) = 2x + 16$ and $x = 4$, then:

$$(f \circ g)(4) = 2(4) + 16 = 8 + 16 = 24$$

Q: What is the value of $(f \circ g)(4)$?

A: The value of $(f \circ g)(4)$ is 24.

Q: What is the value of $(g \circ f)(4)$?

A: The value of $(g \circ f)(4)$ is 16.

Conclusion

In this article, we have answered some frequently asked questions about the composition of functions. We hope that this Q&A guide has been helpful in understanding the composition of functions.

Key Takeaways

• The composition of functions is a way of combining two or more functions to create a new function.
• The composite function $(f \circ g)(x)$ is defined as $f(g(x))$.
• The composite function $(g \circ f)(x)$ is defined as $g(f(x))$.
• The order of the functions matters in the composition of functions.
• The value of $(f \circ g)(4)$ is 24.
• The value of $(g \circ f)(4)$ is 16.

Further Reading

For further reading on the composition of functions, we recommend the following resources:

• Wikipedia: Composition of Functions
• Khan Academy: Composition of Functions
• Mathway: Composition of Functions