Find The Compositions For The Given Functions:Let F ( X ) = 3 X + 2 F(x) = 3x + 2 F ( X ) = 3 X + 2 And G ( X ) = X 2 − 6 G(x) = X^2 - 6 G ( X ) = X 2 − 6 .(a) ( F ∘ G ) ( X ) = (f \circ G)(x) = ( F ∘ G ) ( X ) = □ \square □ (b) ( G ∘ F ) ( X ) = (g \circ F)(x) = ( G ∘ F ) ( X ) = □ \square □ (c) ( F ∘ G ) ( − 1 ) = (f \circ G)(-1) = ( F ∘ G ) ( − 1 ) =

Introduction

In mathematics, a composition of functions is a way of combining two or more functions to create a new function. This is a fundamental concept in algebra and calculus, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore the composition of functions, with a focus on finding the compositions for the given functions $f(x) = 3x + 2$ and $g(x) = x^2 - 6$.

What is a Composition of Functions?

A 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 Composition of $f$ and $g$

Now, let's find the composition of $f$ and $g$. We have:

$$f(x) = 3x + 2$$

$$g(x) = x^2 - 6$$

To find the composition $(f \circ g)(x)$, we need to substitute $g(x)$ into $f(x)$:

$$(f \circ g)(x) = f(g(x)) = 3(g(x)) + 2$$

$$= 3(x^2 - 6) + 2$$

$$= 3x^2 - 18 + 2$$

$$= 3x^2 - 16$$

Therefore, the composition of $f$ and $g$ is:

$$(f \circ g)(x) = 3x^2 - 16$$

Finding the Composition of $g$ and $f$

Now, let's find the composition of $g$ and $f$. We have:

$$(g \circ f)(x) = g(f(x)) = (f(x))^2 - 6$$

$$= (3x + 2)^2 - 6$$

$$= 9x^2 + 12x + 4 - 6$$

$$= 9x^2 + 12x - 2$$

Therefore, the composition of $g$ and $f$ is:

$$(g \circ f)(x) = 9x^2 + 12x - 2$$

Evaluating the Composition at a Specific Value

Now, let's evaluate the composition $(f \circ g)(x)$ at $x = -1$. We have:

$$(f \circ g)(-1) = 3(-1)^2 - 16$$

$$= 3(1) - 16$$

$$= 3 - 16$$

$$= -13$$

Therefore, the value of the composition $(f \circ g)(x)$ at $x = -1$ is $-13$.

Conclusion

In this article, we have explored the composition of functions, with a focus on finding the compositions for the given functions $f(x) = 3x + 2$ and $g(x) = x^2 - 6$. We have shown how to find the composition of two functions, and how to evaluate the composition at a specific value. The composition of functions is a fundamental concept in mathematics, and it has numerous applications in various fields. We hope that this article has provided a comprehensive guide to the composition of functions.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Economists" by Carl P. Simon and Lawrence Blume

Further Reading

• [1] "Introduction to Algebra" by Richard Rusczyk
• [2] "Calculus: Early Transcendentals" by James Stewart
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Composition of Functions: A Q&A Guide =====================================

Introduction

In our previous article, we explored the composition of functions, with a focus on finding the compositions for the given functions $f(x) = 3x + 2$ and $g(x) = x^2 - 6$. In this article, we will provide a Q&A guide to help you understand the composition of functions better.

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 composition of two functions?

A: To find the composition of two functions, you need to substitute the second function into the first function. For example, if we have:

$$f(x) = 3x + 2$$

$$g(x) = x^2 - 6$$

To find the composition $(f \circ g)(x)$, we need to substitute $g(x)$ into $f(x)$:

$$(f \circ g)(x) = f(g(x)) = 3(g(x)) + 2$$

$$= 3(x^2 - 6) + 2$$

$$= 3x^2 - 18 + 2$$

$$= 3x^2 - 16$$

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

A: The composition $(f \circ g)(x)$ is different from $(g \circ f)(x)$. To find $(g \circ f)(x)$, we need to substitute $f(x)$ into $g(x)$:

$$(g \circ f)(x) = g(f(x)) = (f(x))^2 - 6$$

$$= (3x + 2)^2 - 6$$

$$= 9x^2 + 12x + 4 - 6$$

$$= 9x^2 + 12x - 2$$

Q: How do I evaluate the composition of functions at a specific value?

A: To evaluate the composition of functions at a specific value, you need to substitute the value into the composition. For example, if we have:

$$(f \circ g)(x) = 3x^2 - 16$$

To evaluate $(f \circ g)(x)$ at $x = -1$, we need to substitute $x = -1$ into the composition:

$$(f \circ g)(-1) = 3(-1)^2 - 16$$

$$= 3(1) - 16$$

$$= 3 - 16$$

$$= -13$$

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

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

• Not following the order of operations
• Not substituting the correct function into the other function
• Not simplifying the expression correctly
• Not evaluating the composition at the correct value

Q: What are some real-world applications of compositions of functions?

A: Compositions of functions have numerous real-world applications, including:

• Physics: Compositions of functions are used to model the motion of objects under the influence of gravity and other forces.
• Engineering: Compositions of functions are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Compositions of functions are used to model economic systems and make predictions about future economic trends.

Conclusion

In this article, we have provided a Q&A guide to help you understand the composition of functions better. We have covered topics such as finding the composition of two functions, evaluating the composition at a specific value, and avoiding common mistakes. We hope that this article has been helpful in your understanding of compositions of functions.