Refer To The Functions { R, P $}$, And { Q $}$.Find The Function { \left(\frac{q}{p}\right)(x)$}$ And Write The Domain In Interval Notation.Given:${ R(x) = -5x }$ { P(x) = X^2 + 4x \} $[ Q(x) =

Introduction

In mathematics, function composition is a fundamental concept that involves combining two or more functions to create a new function. Given two functions, ${ p(x) }$ and ${ q(x) }$, the composition of these functions is denoted as ${ (p \circ q)(x) }$ or ${ p(q(x)) }$. In this article, we will explore the composition of the functions ${ r(x) = -5x }$, ${ p(x) = x^2 + 4x }$, and ${ q(x) = x^3 - 2x^2 + 3x - 1 }$ to find the function ${ \left(\frac{q}{p}\right)(x) }$ and analyze its domain in interval notation.

Function Composition

To find the composition of the functions ${ r(x) }$, ${ p(x) }$, and ${ q(x) }$, we need to understand the concept of function composition. The composition of two functions ${ f(x) }$ and ${ g(x) }$ is denoted as ${ (f \circ g)(x) }$ or ${ f(g(x)) }$. This means that we need to plug in the function ${ g(x) }$ into the function ${ f(x) }$ to get the resulting function.

In this case, we are given the functions ${ r(x) = -5x }$, ${ p(x) = x^2 + 4x }$, and ${ q(x) = x^3 - 2x^2 + 3x - 1 }$. To find the composition of these functions, we need to plug in the functions ${ p(x) }$ and ${ q(x) }$ into the function ${ r(x) }$.

Finding the Composition

To find the composition of the functions ${ r(x) }$, ${ p(x) }$, and ${ q(x) }$, we need to plug in the functions ${ p(x) }$ and ${ q(x) }$ into the function ${ r(x) }$. This means that we need to replace the variable ${ x }$ in the function ${ r(x) }$ with the functions ${ p(x) }$ and ${ q(x) }$.

Let's start by finding the composition of the functions ${ r(x) }$ and ${ p(x) }$. We can do this by plugging in the function ${ p(x) }$ into the function ${ r(x) }$.

${ r(p(x)) = -5(p(x)) }$

Now, we can substitute the function ${ p(x) = x^2 + 4x }$ into the equation above.

${ r(p(x)) = -5(x^2 + 4x) }$

Expanding the equation above, we get:

${ r(p(x)) = -5x^2 - 20x }$

Now, let's find the composition of the functions ${ r(x) }$ and ${ q(x) }$. We can do this by plugging in the function ${ q(x) }$ into the function ${ r(x) }$.

${ r(q(x)) = -5(q(x)) }$

Now, we can substitute the function ${ q(x) = x^3 - 2x^2 + 3x - 1 }$ into the equation above.

${ r(q(x)) = -5(x^3 - 2x^2 + 3x - 1) }$

Expanding the equation above, we get:

${ r(q(x)) = -5x^3 + 10x^2 - 15x + 5 }$

Finding the Function ${ \left(\frac{q}{p}\right)(x) }$

To find the function ${ \left(\frac{q}{p}\right)(x) }$, we need to divide the function ${ q(x) }$ by the function ${ p(x) }$.

Let's start by finding the quotient of the functions ${ q(x) }$ and ${ p(x) }$. We can do this by dividing the function ${ q(x) }$ by the function ${ p(x) }$.

${ \left(\frac{q}{p}\right)(x) = \frac{q(x)}{p(x)} }$

Now, we can substitute the functions ${ q(x) = x^3 - 2x^2 + 3x - 1 }$ and ${ p(x) = x^2 + 4x }$ into the equation above.

${ \left(\frac{q}{p}\right)(x) = \frac{x^3 - 2x^2 + 3x - 1}{x^2 + 4x} }$

To simplify the equation above, we can factor the numerator and denominator.

${ \left(\frac{q}{p}\right)(x) = \frac{(x - 1)(x^2 + x - 1)}{(x + 4)(x + 1)} }$

Analyzing the Domain

To analyze the domain of the function ${ \left(\frac{q}{p}\right)(x) }$, we need to find the values of ${ x }$ that make the denominator equal to zero.

Let's start by finding the values of ${ x }$ that make the denominator ${ (x + 4)(x + 1) }$ equal to zero.

${ (x + 4)(x + 1) = 0 }$

This means that either ${ x + 4 = 0 }$ or ${ x + 1 = 0 }$.

Solving for ${ x }$, we get:

${ x + 4 = 0 \implies x = -4 }$

${ x + 1 = 0 \implies x = -1 }$

Therefore, the values of ${ x }$ that make the denominator equal to zero are ${ x = -4 }$ and ${ x = -1 }$.

To find the domain of the function ${ \left(\frac{q}{p}\right)(x) }$, we need to exclude the values of ${ x }$ that make the denominator equal to zero.

Therefore, the domain of the function ${ \left(\frac{q}{p}\right)(x) }$ is:

${ (-\infty, -4) \cup (-4, -1) \cup (-1, \infty) }$

Conclusion

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Q: What is function composition?

A: Function composition is a fundamental concept in mathematics that involves combining two or more functions to create a new function. Given two functions, ${ p(x) }$ and ${ q(x) }$, the composition of these functions is denoted as ${ (p \circ q)(x) }$ or ${ p(q(x)) }$.

Q: How do I find the composition of two functions?

A: To find the composition of two functions, you need to plug in the second function into the first function. For example, if we have the functions ${ p(x) = x^2 + 4x }$ and ${ q(x) = x^3 - 2x^2 + 3x - 1 }$, we can find the composition of these functions by plugging in the function ${ q(x) }$ into the function ${ p(x) }$.

Q: What is the difference between ${ (p \circ q)(x) }$ and ${ p(q(x)) }$?

A: ${ (p \circ q)(x) }$ and ${ p(q(x)) }$ are two notations for the composition of two functions. They are equivalent and represent the same function.

Q: How do I find the domain of a function?

A: To find the domain of a function, you need to find the values of ${ x }$ that make the denominator equal to zero. If the denominator is a polynomial, you can find the values of ${ x }$ that make the polynomial equal to zero by factoring or using the quadratic formula.

Q: What is the domain of the function ${ \left(\frac{q}{p}\right)(x) }$?

A: The domain of the function ${ \left(\frac{q}{p}\right)(x) }$ is ${ (-\infty, -4) \cup (-4, -1) \cup (-1, \infty) }$. This means that the function is defined for all real numbers except ${ x = -4 }$ and ${ x = -1 }$.

Q: Can I simplify the function ${ \left(\frac{q}{p}\right)(x) }$?

A: Yes, you can simplify the function ${ \left(\frac{q}{p}\right)(x) }$ by factoring the numerator and denominator. This can help you to better understand the function and its behavior.

Q: How do I use function composition in real-world applications?

A: Function composition is a powerful tool that can be used in a variety of real-world applications, such as:

• Modeling population growth and decline
• Analyzing financial data and predicting stock prices
• Understanding the behavior of complex systems, such as electrical circuits and mechanical systems
• Developing algorithms and models for machine learning and artificial intelligence

Q: What are some common mistakes to avoid when working with function composition?

A: Some common mistakes to avoid when working with function composition include:

• Not checking the domain of the function
• Not simplifying the function
• Not using the correct notation for function composition
• Not understanding the behavior of the function

Q: Where can I learn more about function composition and domain analysis?

A: There are many resources available to learn more about function composition and domain analysis, including:

• Online tutorials and videos
• Textbooks and reference books
• Online courses and degree programs
• Professional conferences and workshops

By following these resources and practicing with examples, you can develop a deeper understanding of function composition and domain analysis and apply these concepts to real-world problems.