Refer To The Functions { R $}$ And { P $}$. Find The Function { (p-r)(x)$}$ And Write The Domain In Interval Notation.Given:${ R(x) = -5x }$ { P(x) = X^2 - 2x \} $[ Q(x) = \sqrt{7-x}

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In this article, we will explore the concept of function composition and analyze the domain of a given function. We will start by defining two functions, r(x)=−5x r(x) = -5x } and p(x)=x2−2x{ p(x) = x^2 - 2x , and then find the function (p−r)(x){ (p-r)(x) }. Finally, we will determine the domain of the resulting function in interval notation.

Function Composition

Function composition is a process of combining two or more functions to create a new function. Given two functions, f(x){ f(x) } and g(x){ g(x) }, the composition of f{ f } and g{ g } is defined as (f∘g)(x)=f(g(x)){ (f \circ g)(x) = f(g(x)) }. In other words, we first apply the function g{ g } to the input x{ x }, and then apply the function f{ f } to the result.

Finding the Function (p−r)(x){ (p-r)(x) }

To find the function (p−r)(x){ (p-r)(x) }, we need to subtract the function r(x){ r(x) } from the function p(x){ p(x) }. This can be done by applying the subtraction operation to the corresponding terms of the two functions.

(p−r)(x)=p(x)−r(x){ (p-r)(x) = p(x) - r(x) } =(x2−2x)−(−5x){ = (x^2 - 2x) - (-5x) } =x2−2x+5x{ = x^2 - 2x + 5x } =x2+3x{ = x^2 + 3x }

Therefore, the function (p−r)(x){ (p-r)(x) } is given by x2+3x{ x^2 + 3x }.

Domain Analysis

To determine the domain of the function (p−r)(x){ (p-r)(x) }, we need to consider the restrictions imposed by the square root function in the original function q(x)=7−x{ q(x) = \sqrt{7-x} }. Since the square root function is only defined for non-negative values, we must ensure that the expression inside the square root is non-negative.

7−x≥0{ 7 - x \geq 0 } x≤7{ x \leq 7 }

This means that the domain of the function q(x){ q(x) } is (−∞,7]{ (-\infty, 7] }. However, since the function (p−r)(x){ (p-r)(x) } is a polynomial function, it is defined for all real numbers. Therefore, the domain of the function (p−r)(x){ (p-r)(x) } is the entire set of real numbers, which can be represented in interval notation as (−∞,∞){ (-\infty, \infty) }.

Conclusion

In this article, we have explored the concept of function composition and analyzed the domain of a given function. We have found the function (p−r)(x){ (p-r)(x) } by subtracting the function r(x){ r(x) } from the function p(x){ p(x) }, and determined the domain of the resulting function in interval notation. The domain of the function (p−r)(x){ (p-r)(x) } is the entire set of real numbers, represented in interval notation as (−∞,∞){ (-\infty, \infty) }.

References

Discussion

What are some other examples of function composition? How can we use function composition to solve real-world problems? Share your thoughts and ideas in the comments below!

Related Topics

  • Function Composition
  • Domain of a Function
  • Polynomial Functions
  • Square Root Functions

Keywords

  • Function Composition
  • Domain Analysis
  • Polynomial Functions
  • Square Root Functions
  • Interval Notation
    Q&A: Function Composition and Domain Analysis =====================================================

In our previous article, we explored the concept of function composition and analyzed the domain of a given function. In this article, we will answer some frequently asked questions related to function composition and domain analysis.

Q: What is function composition?

A: Function composition is a process of combining two or more functions to create a new function. Given two functions, f(x){ f(x) } and g(x){ g(x) }, the composition of f{ f } and g{ g } is defined as (f∘g)(x)=f(g(x)){ (f \circ g)(x) = f(g(x)) }. In other words, we first apply the function g{ g } to the input x{ x }, and then apply the function f{ f } to the result.

Q: How do I find the function (p−r)(x){ (p-r)(x) }?

A: To find the function (p−r)(x){ (p-r)(x) }, you need to subtract the function r(x){ r(x) } from the function p(x){ p(x) }. This can be done by applying the subtraction operation to the corresponding terms of the two functions.

(p−r)(x)=p(x)−r(x){ (p-r)(x) = p(x) - r(x) } =(x2−2x)−(−5x){ = (x^2 - 2x) - (-5x) } =x2−2x+5x{ = x^2 - 2x + 5x } =x2+3x{ = x^2 + 3x }

Q: What is the domain of the function (p−r)(x){ (p-r)(x) }?

A: The domain of the function (p−r)(x){ (p-r)(x) } is the entire set of real numbers, represented in interval notation as (−∞,∞){ (-\infty, \infty) }. This is because the function (p−r)(x){ (p-r)(x) } is a polynomial function, which is defined for all real numbers.

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

A: To determine the domain of a function, you need to consider the restrictions imposed by the function. For example, if the function contains a square root, you need to ensure that the expression inside the square root is non-negative. If the function contains a fraction, you need to ensure that the denominator is not equal to zero.

Q: What are some examples of function composition?

A: Here are a few examples of function composition:

  • (f∘g)(x)=f(g(x))=2(3x)=6x{ (f \circ g)(x) = f(g(x)) = 2(3x) = 6x }
  • (f∘g)(x)=f(g(x))=sin⁡(cos⁡(x)){ (f \circ g)(x) = f(g(x)) = \sin(\cos(x)) }
  • (f∘g)(x)=f(g(x))=1−x2{ (f \circ g)(x) = f(g(x)) = \sqrt{1-x^2} }

Q: How can I use function composition to solve real-world problems?

A: Function composition can be used to solve a wide range of real-world problems. For example, you can use function composition to model population growth, predict stock prices, or analyze the behavior of complex systems.

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

A: Here are a few common mistakes to avoid when working with function composition:

  • Not following the order of operations
  • Not simplifying the expression
  • Not considering the domain of the function
  • Not using the correct notation

Conclusion

In this article, we have answered some frequently asked questions related to function composition and domain analysis. We hope that this article has been helpful in clarifying some of the concepts and providing a better understanding of function composition and domain analysis.

References

Discussion

Do you have any questions or topics you would like to discuss related to function composition and domain analysis? Share your thoughts and ideas in the comments below!

Related Topics

  • Function Composition
  • Domain of a Function
  • Polynomial Functions
  • Square Root Functions
  • Interval Notation

Keywords

  • Function Composition
  • Domain Analysis
  • Polynomial Functions
  • Square Root Functions
  • Interval Notation
  • Real-World Applications