For F ( X ) = X + 5 F(x) = X + 5 F ( X ) = X + 5 And G ( X ) = 5 X + 4 G(x) = 5x + 4 G ( X ) = 5 X + 4 , 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 ) ( − 2 (f \circ G)(-2 ( F ∘ G ) ( − 2 ]d. ( G ∘ F ) ( − 2 (g \circ F)(-2 ( G ∘ F ) ( − 2 ]

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Introduction

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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 branches of mathematics, including algebra, calculus, and analysis. In this article, we will explore the composition of functions, focusing on the given functions $f(x) = x + 5$ and $g(x) = 5x + 4$. We will find the composite functions $(f \circ g)(x)$ and $(g \circ f)(x)$, as well as their values at $x = -2$.

Composition of Functions: Definition and Notation

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The composition of two functions $f$ 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

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

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

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

Now, we substitute $5x + 4$ into the function $f(x) = x + 5$.

$$f(5x + 4) = (5x + 4) + 5$$

Simplifying the expression, we get:

$$(f \circ g)(x) = 5x + 9$$

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

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

$$g(f(x)) = g(x + 5)$$

Now, we substitute $x + 5$ into the function $g(x) = 5x + 4$.

$$g(x + 5) = 5(x + 5) + 4$$

Simplifying the expression, we get:

$$(g \circ f)(x) = 5x + 29$$

Evaluating the Composite Functions at $x = -2$

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c. $(f \circ g)(-2)$

To evaluate the composite function $(f \circ g)(x)$ at $x = -2$, we substitute $-2$ into the expression for $(f \circ g)(x)$.

$$(f \circ g)(-2) = 5(-2) + 9$$

Simplifying the expression, we get:

$$(f \circ g)(-2) = -10 + 9 = -1$$

d. $(g \circ f)(-2)$

To evaluate the composite function $(g \circ f)(x)$ at $x = -2$, we substitute $-2$ into the expression for $(g \circ f)(x)$.

$$(g \circ f)(-2) = 5(-2) + 29$$

Simplifying the expression, we get:

$$(g \circ f)(-2) = -10 + 29 = 19$$

Conclusion

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In this article, we have explored the composition of functions, focusing on the given functions $f(x) = x + 5$ and $g(x) = 5x + 4$. We have found the composite functions $(f \circ g)(x)$ and $(g \circ f)(x)$, as well as their values at $x = -2$. The composition of functions is a powerful tool in mathematics, allowing us to create new functions from existing ones. We hope that this article has provided a comprehensive guide to the composition of functions.

Future Directions

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The composition of functions has numerous applications in various branches of mathematics, including algebra, calculus, and analysis. Some potential future directions for research in this area include:

• Composition of functions with multiple variables: Extending the concept of composition of functions to functions with multiple variables.
• Composition of functions with different domains and ranges: Investigating the composition of functions with different domains and ranges.
• Composition of functions with non-linear transformations: Studying the composition of functions with non-linear transformations.

By exploring these and other directions, we can continue to deepen our understanding of the composition of functions and its applications in mathematics.

References

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• [1] "Composition of Functions" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f7f7f/composition-of-functions
• [2] "Composition of Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/composition.html

Glossary

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• Composition of functions: The process of combining two or more functions to create a new function.
• Domain: The set of all possible input values for a function.
• Range: The set of all possible output values for a function.
• Function: A relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
  

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Q&A: Composition of Functions

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Q: What is the composition of functions?

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A: The composition of functions is the process of combining two or more functions to create a new function. This 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 $(f \circ g)(x)$?

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A: To find the composite function $(f \circ g)(x)$, you need to substitute $g(x)$ into $f(x)$. For example, if $f(x) = x + 5$ and $g(x) = 5x + 4$, then:

$$(f \circ g)(x) = f(g(x)) = f(5x + 4) = (5x + 4) + 5 = 5x + 9$$

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

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A: To find the composite function $(g \circ f)(x)$, you need to substitute $f(x)$ into $g(x)$. For example, if $f(x) = x + 5$ and $g(x) = 5x + 4$, then:

$$(g \circ f)(x) = g(f(x)) = g(x + 5) = 5(x + 5) + 4 = 5x + 29$$

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

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A: The main difference between $(f \circ g)(x)$ and $(g \circ f)(x)$ is the order in which the functions are applied. In $(f \circ g)(x)$, the function $g$ is applied first, followed by the function $f$. In $(g \circ f)(x)$, the function $f$ is applied first, followed by the function $g$.

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

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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 expression for $(f \circ g)(x)$. For example, if $(f \circ g)(x) = 5x + 9$ and we want to evaluate it at $x = -2$, then:

$$(f \circ g)(-2) = 5(-2) + 9 = -10 + 9 = -1$$

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

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A: To evaluate the composite function $(g \circ f)(x)$ at a specific value of $x$, you need to substitute the value of $x$ into the expression for $(g \circ f)(x)$. For example, if $(g \circ f)(x) = 5x + 29$ and we want to evaluate it at $x = -2$, then:

$$(g \circ f)(-2) = 5(-2) + 29 = -10 + 29 = 19$$

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

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A: The composition of functions has numerous applications in various branches of mathematics, including algebra, calculus, and analysis. Some common applications include:

• Modeling real-world phenomena: The composition of functions can be used to model real-world phenomena, such as population growth, chemical reactions, and electrical circuits.
• Solving equations: The composition of functions can be used to solve equations, such as quadratic equations and systems of equations.
• Optimization: The composition of functions can be used to optimize functions, such as finding the maximum or minimum value of a function.

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

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

• Confusing the order of the functions: Make sure to keep track of the order in which the functions are applied.
• Not simplifying the expression: Make sure to simplify the expression for the composite function.
• Not evaluating the function at the correct value of $x$: Make sure to evaluate the function at the correct value of $x$.

By following these tips and avoiding common mistakes, you can master the composition of functions and apply it to a wide range of mathematical problems.