Find { (q \circ R)(-1)$} . . . { \begin{array}{c} q(x) = 2x \\ r(x) = X^2 + 5x \\ (q \circ R)(-1) = \square \end{array} \}

Introduction

In mathematics, the composition of functions is a fundamental concept that plays a crucial role in various branches of mathematics, including algebra, calculus, and analysis. The composition of functions is a way of combining two or more functions to create a new function. In this article, we will focus on finding the value of {(q \circ r)(-1)$}$, where $q(x) = 2x$ and $r(x) = x^2 + 5x$. We will use the concept of composition of functions to evaluate this expression and provide a step-by-step solution.

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 $f \circ g$ and is defined as:

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

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

Evaluating {(q \circ r)(-1)$}$

To evaluate {(q \circ r)(-1)$}$, we need to find the value of $r(-1)$ first, and then plug this value into the function $q(x) = 2x$.

Step 1: Evaluate $r(-1)$

We are given that $r(x) = x^2 + 5x$. To evaluate $r(-1)$, we substitute $x = -1$ into the function:

${r(-1) = (-1)^2 + 5(-1)}$ ${r(-1) = 1 - 5}$ ${r(-1) = -4}$

Step 2: Evaluate $q(r(-1))$

Now that we have found the value of $r(-1)$, we can plug this value into the function $q(x) = 2x$:

${q(r(-1)) = q(-4)}$ ${q(r(-1)) = 2(-4)}$ ${q(r(-1)) = -8}$

Conclusion

In this article, we have evaluated the expression {(q \circ r)(-1)$}$ using the concept of composition of functions. We first found the value of $r(-1)$, and then plugged this value into the function $q(x) = 2x$. The final answer is $\boxed{-8}$.

Frequently Asked Questions

• What is the composition of functions? The composition of functions is a way of combining two or more functions to create a new function.
• How do you evaluate the composition of functions? To evaluate the composition of functions, you need to apply the inner function to the input, and then apply the outer function to the result.
• What is the value of {(q \circ r)(-1)$}$? The value of {(q \circ r)(-1)$}$ is $\boxed{-8}$.

Further Reading

• Composition of Functions: A Comprehensive Guide
• Evaluating Functions: A Step-by-Step Guide
• Algebra: A Beginner's Guide

References

• [1] "Composition of Functions" by Khan Academy
• [2] "Evaluating Functions" by Mathway
• [3] "Algebra" by OpenStax
  

Introduction

In our previous article, we explored the concept of composition of functions and evaluated the expression {(q \circ r)(-1)$}$. In this article, we will address some of the most frequently asked questions about composition of functions. Whether you are a student, a teacher, or simply someone interested in mathematics, this article will provide you with a comprehensive guide to composition of functions.

Q&A: Composition of Functions

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 $f \circ g$ and is defined as:

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

Q: How do you evaluate the composition of functions?

A: To evaluate the composition of functions, you need to apply the inner function to the input, and then apply the outer function to the result. For example, if we have the composition of functions $f \circ g$, we first evaluate $g(x)$, and then plug this value into the function $f(x)$.

Q: What is the difference between composition of functions and function notation?

A: Composition of functions and function notation are two different concepts in mathematics. Function notation is a way of representing a function as a mathematical expression, while composition of functions is a way of combining two or more functions to create a new function.

Q: Can you provide an example of composition of functions?

A: Yes, let's consider an example. Suppose we have two functions $f(x) = 2x$ and $g(x) = x^2 + 5x$. We can evaluate the composition of functions $f \circ g$ as follows:

${(f \circ g)(x) = f(g(x))}$ ${(f \circ g)(x) = f(x^2 + 5x)}$ ${(f \circ g)(x) = 2(x^2 + 5x)}$ ${(f \circ g)(x) = 2x^2 + 10x}$

Q: What is the value of {(q \circ r)(-1)$}$?

A: The value of {(q \circ r)(-1)$}$ is $\boxed{-8}$.

Q: Can you provide a step-by-step solution to evaluating the composition of functions?

A: Yes, let's consider an example. Suppose we have two functions $f(x) = 2x$ and $g(x) = x^2 + 5x$. We can evaluate the composition of functions $f \circ g$ as follows:

1. Evaluate $g(-1)$: ${g(-1) = (-1)^2 + 5(-1)}$ ${g(-1) = 1 - 5}$ ${g(-1) = -4}$
2. Evaluate $f(g(-1))$: ${f(g(-1)) = f(-4)}$ ${f(g(-1)) = 2(-4)}$ ${f(g(-1)) = -8}$

Conclusion

In this article, we have addressed some of the most frequently asked questions about composition of functions. Whether you are a student, a teacher, or simply someone interested in mathematics, this article will provide you with a comprehensive guide to composition of functions.

Frequently Asked Questions

• What is the composition of functions?
• How do you evaluate the composition of functions?
• What is the difference between composition of functions and function notation?
• Can you provide an example of composition of functions?
• What is the value of {(q \circ r)(-1)$}$?
• Can you provide a step-by-step solution to evaluating the composition of functions?

Further Reading

• Composition of Functions: A Comprehensive Guide
• Evaluating Functions: A Step-by-Step Guide
• Algebra: A Beginner's Guide

References

• [1] "Composition of Functions" by Khan Academy
• [2] "Evaluating Functions" by Mathway
• [3] "Algebra" by OpenStax