Suppose That The Functions $q$ And $r$ Are Defined As Follows:$ \begin{align*} q(x) &= 3x - 5 \\ r(x) &= 2x - 3 \end{align*} $Find The Following:$ \begin{align*} (q \circ R)(-1) &= \\ (r \circ Q)(-1)

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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 branches of mathematics, including algebra, calculus, and analysis. In this article, we will explore the composition of functions, specifically the functions qq and rr, which are defined as follows:

Function Definitions

\begin{align*} q(x) &= 3x - 5 \ r(x) &= 2x - 3 \end{align*}

We will find the composition of these functions, denoted as (q∘r)(x)(q \circ r)(x) and (r∘q)(x)(r \circ q)(x), and evaluate them at x=−1x = -1.

Composition of Functions

The composition of two functions ff and gg is denoted as (f∘g)(x)(f \circ g)(x) and is defined as:

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

In our case, we have:

(q∘r)(x)=q(r(x))(q \circ r)(x) = q(r(x))

Substituting the definition of r(x)r(x) into the equation above, we get:

(q∘r)(x)=q(2x−3)(q \circ r)(x) = q(2x - 3)

Now, we substitute the definition of q(x)q(x) into the equation above:

(q∘r)(x)=3(2x−3)−5(q \circ r)(x) = 3(2x - 3) - 5

Simplifying the equation, we get:

(q∘r)(x)=6x−9−5(q \circ r)(x) = 6x - 9 - 5

(q∘r)(x)=6x−14(q \circ r)(x) = 6x - 14

Similarly, we can find the composition (r∘q)(x)(r \circ q)(x):

(r∘q)(x)=r(q(x))(r \circ q)(x) = r(q(x))

Substituting the definition of q(x)q(x) into the equation above, we get:

(r∘q)(x)=r(3x−5)(r \circ q)(x) = r(3x - 5)

Now, we substitute the definition of r(x)r(x) into the equation above:

(r∘q)(x)=2(3x−5)−3(r \circ q)(x) = 2(3x - 5) - 3

Simplifying the equation, we get:

(r∘q)(x)=6x−10−3(r \circ q)(x) = 6x - 10 - 3

(r∘q)(x)=6x−13(r \circ q)(x) = 6x - 13

Evaluating the Compositions at x=−1x = -1

Now that we have found the compositions (q∘r)(x)(q \circ r)(x) and (r∘q)(x)(r \circ q)(x), we can evaluate them at x=−1x = -1.

For (q∘r)(x)(q \circ r)(x), we substitute x=−1x = -1 into the equation:

(q∘r)(−1)=6(−1)−14(q \circ r)(-1) = 6(-1) - 14

(q∘r)(−1)=−6−14(q \circ r)(-1) = -6 - 14

(q∘r)(−1)=−20(q \circ r)(-1) = -20

For (r∘q)(x)(r \circ q)(x), we substitute x=−1x = -1 into the equation:

(r∘q)(−1)=6(−1)−13(r \circ q)(-1) = 6(-1) - 13

(r∘q)(−1)=−6−13(r \circ q)(-1) = -6 - 13

(r∘q)(−1)=−19(r \circ q)(-1) = -19

Conclusion

In this article, we explored the composition of functions, specifically the functions qq and rr. We found the compositions (q∘r)(x)(q \circ r)(x) and (r∘q)(x)(r \circ q)(x) and evaluated them at x=−1x = -1. The results are:

(q∘r)(−1)=−20(q \circ r)(-1) = -20

(r∘q)(−1)=−19(r \circ q)(-1) = -19

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 branches of mathematics, including algebra, calculus, and analysis.

References

  • [1] "Composition of Functions" by Khan Academy
  • [2] "Composition of Functions" by Math Open Reference
  • [3] "Composition of Functions" by Wolfram MathWorld

Discussion

The composition of functions is a fundamental concept in mathematics. It allows us to combine two or more functions to create a new function. In this article, we explored the composition of the functions qq and rr. We found the compositions (q∘r)(x)(q \circ r)(x) and (r∘q)(x)(r \circ q)(x) and evaluated them at x=−1x = -1. The results are:

(q∘r)(−1)=−20(q \circ r)(-1) = -20

(r∘q)(−1)=−19(r \circ q)(-1) = -19

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 branches of mathematics, including algebra, calculus, and analysis.

Related Topics

Introduction

In our previous article, we explored the composition of functions, specifically the functions qq and rr. We found the compositions (q∘r)(x)(q \circ r)(x) and (r∘q)(x)(r \circ q)(x) and evaluated them at x=−1x = -1. 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 two functions ff and gg is denoted as (f∘g)(x)(f \circ g)(x) and is defined as:

(f∘g)(x)=f(g(x))(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 definition of one function into the other function. For example, to find (q∘r)(x)(q \circ r)(x), you would substitute the definition of r(x)r(x) into the definition of q(x)q(x).

Q: What is the difference between (q∘r)(x)(q \circ r)(x) and (r∘q)(x)(r \circ q)(x)?

A: The composition (q∘r)(x)(q \circ r)(x) is different from (r∘q)(x)(r \circ q)(x) because the order of the functions is reversed. In (q∘r)(x)(q \circ r)(x), the function r(x)r(x) is evaluated first, and then the result is passed to the function q(x)q(x). In (r∘q)(x)(r \circ q)(x), the function q(x)q(x) is evaluated first, and then the result is passed to the function r(x)r(x).

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

A: To evaluate the composition of functions at a specific value of xx, you need to substitute the value of xx into the composition of functions. For example, to evaluate (q∘r)(−1)(q \circ r)(-1), you would substitute x=−1x = -1 into the composition (q∘r)(x)(q \circ r)(x).

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

A: The composition of functions has many applications in mathematics and science. Some common applications include:

  • Modeling real-world phenomena, such as population growth or chemical reactions
  • Solving systems of equations
  • Finding the inverse of a function
  • Evaluating the derivative of a function

Q: Can I use the composition of functions to find the inverse of a function?

A: Yes, you can use the composition of functions to find the inverse of a function. If you have a function f(x)f(x) and you want to find its inverse, you can use the composition (f−1∘f)(x)=x(f^{-1} \circ f)(x) = x to find the inverse.

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

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

  • Not following the order of operations
  • Not substituting the definition of one function into the other function
  • Not evaluating the composition of functions at a specific value of xx

Conclusion

In this article, we answered some frequently asked questions about the composition of functions. We hope that this guide has been helpful in understanding the composition of functions and how to apply it in various mathematical and scientific contexts.

References

  • [1] "Composition of Functions" by Khan Academy
  • [2] "Composition of Functions" by Math Open Reference
  • [3] "Composition of Functions" by Wolfram MathWorld

Discussion

The composition of functions is a powerful tool in mathematics and science. It allows us to combine two or more functions to create a new function. In this article, we answered some frequently asked questions about the composition of functions and provided some common applications and mistakes to avoid.

Related Topics