Suppose That The Functions Are Given As Follows:${ U(x) = -4x + 5 }$ { W(x) = -2x + 2 \} Find The Following:${ (w \cdot U)(3) = }$ { (u \cdot W)(3) = \}

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Introduction

In mathematics, functions play a crucial role in various mathematical operations. Given two functions, u(x) and w(x), we can perform various operations such as addition, subtraction, multiplication, and division. In this article, we will focus on finding the product of two functions, u(x) and w(x), at a specific point, x = 3.

Given Functions

The given functions are:

u(x)=−4x+5{ u(x) = -4x + 5 }

w(x)=−2x+2{ w(x) = -2x + 2 }

Finding the Product of Functions

To find the product of two functions, u(x) and w(x), we can use the following formula:

(uâ‹…w)(x)=u(x)â‹…w(x){ (u \cdot w)(x) = u(x) \cdot w(x) }

Substituting the given functions, we get:

(u⋅w)(x)=(−4x+5)⋅(−2x+2){ (u \cdot w)(x) = (-4x + 5) \cdot (-2x + 2) }

Finding (w \cdot u)(3)

To find (w \cdot u)(3), we need to substitute x = 3 into the product of functions, u(x) and w(x).

(w⋅u)(3)=(−2⋅3+2)⋅(−4⋅3+5){ (w \cdot u)(3) = (-2 \cdot 3 + 2) \cdot (-4 \cdot 3 + 5) }

(w⋅u)(3)=(−6+2)⋅(−12+5){ (w \cdot u)(3) = (-6 + 2) \cdot (-12 + 5) }

(w⋅u)(3)=−4⋅−7{ (w \cdot u)(3) = -4 \cdot -7 }

(wâ‹…u)(3)=28{ (w \cdot u)(3) = 28 }

Finding (u \cdot w)(3)

To find (u \cdot w)(3), we need to substitute x = 3 into the product of functions, u(x) and w(x).

(u⋅w)(3)=(−4⋅3+5)⋅(−2⋅3+2){ (u \cdot w)(3) = (-4 \cdot 3 + 5) \cdot (-2 \cdot 3 + 2) }

(u⋅w)(3)=(−12+5)⋅(−6+2){ (u \cdot w)(3) = (-12 + 5) \cdot (-6 + 2) }

(u⋅w)(3)=−7⋅−4{ (u \cdot w)(3) = -7 \cdot -4 }

(uâ‹…w)(3)=28{ (u \cdot w)(3) = 28 }

Conclusion

In this article, we have found the product of two functions, u(x) and w(x), at a specific point, x = 3. We have also shown that the product of functions is commutative, meaning that the order of the functions does not affect the result.

Key Takeaways

  • The product of two functions, u(x) and w(x), is given by the formula: (u \cdot w)(x) = u(x) \cdot w(x)
  • To find the product of functions at a specific point, x = a, we need to substitute x = a into the product of functions.
  • The product of functions is commutative, meaning that the order of the functions does not affect the result.

Further Reading

If you want to learn more about functions and their properties, I recommend checking out the following resources:

  • Khan Academy: Functions
  • MIT OpenCourseWare: Calculus
  • Wolfram MathWorld: Functions

References

Q&A: Functions and Their Properties

Q: What is the product of two functions?

A: The product of two functions, u(x) and w(x), is given by the formula: (u \cdot w)(x) = u(x) \cdot w(x). This means that we multiply the two functions together, point by point, to get the resulting function.

Q: How do I find the product of two functions at a specific point?

A: To find the product of two functions at a specific point, x = a, we need to substitute x = a into the product of functions. For example, if we want to find the product of u(x) and w(x) at x = 3, we would substitute x = 3 into the formula: (u \cdot w)(3) = u(3) \cdot w(3).

Q: Is the product of functions commutative?

A: Yes, the product of functions is commutative. This means that the order of the functions does not affect the result. For example, (u \cdot w)(x) = (w \cdot u)(x).

Q: What is the difference between the product and the sum of functions?

A: The product of two functions, u(x) and w(x), is given by the formula: (u \cdot w)(x) = u(x) \cdot w(x). The sum of two functions, u(x) and w(x), is given by the formula: (u + w)(x) = u(x) + w(x). The product and sum of functions are two different operations, and they have different properties.

Q: Can I find the product of more than two functions?

A: Yes, you can find the product of more than two functions. For example, if we have three functions, u(x), v(x), and w(x), we can find the product of all three functions by multiplying them together: (u \cdot v \cdot w)(x) = u(x) \cdot v(x) \cdot w(x).

Q: How do I use the product of functions in real-world applications?

A: The product of functions has many real-world applications, such as:

  • Modeling population growth: The product of two functions can be used to model the growth of a population over time.
  • Analyzing financial data: The product of two functions can be used to analyze financial data, such as stock prices and interest rates.
  • Solving optimization problems: The product of two functions can be used to solve optimization problems, such as finding the maximum or minimum of a function.

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

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

  • Not checking the domain of the function before evaluating it.
  • Not checking the range of the function before evaluating it.
  • Not using the correct formula for the product of functions.
  • Not simplifying the expression before evaluating it.

Conclusion

In this article, we have answered some common questions about functions and their properties. We have also discussed the product of functions and its applications in real-world scenarios. We hope that this article has been helpful in clarifying any doubts you may have had about functions.

Key Takeaways

  • The product of two functions, u(x) and w(x), is given by the formula: (u \cdot w)(x) = u(x) \cdot w(x)
  • To find the product of functions at a specific point, x = a, we need to substitute x = a into the product of functions.
  • The product of functions is commutative, meaning that the order of the functions does not affect the result.
  • The product of functions has many real-world applications, such as modeling population growth and analyzing financial data.

Further Reading

If you want to learn more about functions and their properties, I recommend checking out the following resources:

  • Khan Academy: Functions
  • MIT OpenCourseWare: Calculus
  • Wolfram MathWorld: Functions

References