Defining the Product of Functions

Oct 13, 2025 by windailys.com 5 views

In this article, we will explore the concept of finding the range of a product of two functions. We will use the given functions $u(x)=-2x^2+3$ and $v(x)=\frac{1}{x}$ to determine the range of their product, $(u \cdot v)(x)$ . This problem requires a deep understanding of function operations, domain and range, and algebraic manipulation.

Defining the Product of Functions

The product of two functions $u(x)$ and $v(x)$ is defined as $(u \cdot v)(x) = u(x) \cdot v(x)$ . In this case, we have:

(u \cdot v)(x) = (-2x^2+3) \cdot \frac{1}{x}

Simplifying the Product

To simplify the product, we can multiply the two functions together:

(u \cdot v)(x) = \frac{-2x^2+3}{x}

Finding the Domain of the Product

The domain of a function is the set of all possible input values for which the function is defined. In this case, the function is defined for all real numbers except $x=0$ , since division by zero is undefined. Therefore, the domain of the product is:

\text{Domain} = (-\infty, 0) \cup (0, \infty)

Finding the Range of the Product

To find the range of the product, we need to consider the possible output values of the function. Since the function is a rational function, its range will be all real numbers except for the values that make the numerator zero. In this case, the numerator is $-2x^2+3$ , which is a quadratic function. The quadratic function has a minimum value of $3$ at $x=0$ , and it opens downward. Therefore, the range of the product will be all real numbers except for the value $3$ .

Analyzing the Options

Now that we have found the range of the product, we can analyze the options:

A. $\left(\frac{1}{3}, 0\right)$ B. $(3, \infty)$ C. $(-\infty, 3)$ D. $(-\infty,+\infty)$

Since the range of the product is all real numbers except for the value $3$ , the correct answer is:

The Correct Answer is C. $(-\infty, 3)$

Conclusion

In this article, we have explored the concept of finding the range of a product of two functions. We have used the given functions $u(x)=-2x^2+3$ and $v(x)=\frac{1}{x}$ to determine the range of their product, $(u \cdot v)(x)$ . We have found that the range of the product is all real numbers except for the value $3$ . This problem requires a deep understanding of function operations, domain and range, and algebraic manipulation.

Key Takeaways

The product of two functions is defined as $(u \cdot v)(x) = u(x) \cdot v(x)$ .
The domain of a function is the set of all possible input values for which the function is defined.
The range of a function is the set of all possible output values of the function.
A rational function has a range that includes all real numbers except for the values that make the numerator zero.

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Algebra, 2nd edition, Michael Artin

Additional Resources

Khan Academy: Functions and Graphs
MIT OpenCourseWare: Calculus
Wolfram Alpha: Function Operations
Q&A: Finding the Range of a Product of Functions =====================================================

In our previous article, we explored the concept of finding the range of a product of two functions. We used the given functions $u(x)=-2x^2+3$ and $v(x)=\frac{1}{x}$ to determine the range of their product, $(u \cdot v)(x)$ . In this article, we will answer some frequently asked questions related to this topic.

Q: What is the product of two functions?

A: The product of two functions $u(x)$ and $v(x)$ is defined as $(u \cdot v)(x) = u(x) \cdot v(x)$ .

Q: How do I find the domain of a product of functions?

A: To find the domain of a product of functions, you need to consider the domain of each individual function. The product is defined for all input values that are in the domain of both functions.

Q: What is the range of a product of functions?

A: The range of a product of functions is the set of all possible output values of the function. To find the range, you need to consider the possible output values of each individual function and combine them.

Q: How do I simplify a product of functions?

A: To simplify a product of functions, you can multiply the two functions together. This will give you a new function that represents the product of the two original functions.

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values of the function.

Q: Can a function have a range that includes all real numbers?

A: Yes, a function can have a range that includes all real numbers. This is known as a function with a range of $(-\infty, \infty)$ .

Q: How do I determine the range of a rational function?

A: To determine the range of a rational function, you need to consider the possible output values of the function. A rational function has a range that includes all real numbers except for the values that make the numerator zero.

Q: What is the significance of the range of a function?

A: The range of a function is important because it tells us the possible output values of the function. This can be useful in a variety of applications, such as modeling real-world phenomena or solving equations.

Q: Can I use technology to find the range of a function?

A: Yes, you can use technology such as graphing calculators or computer software to find the range of a function. These tools can help you visualize the function and determine its range.

Q: How do I apply the concept of range to real-world problems?

A: The concept of range can be applied to a variety of real-world problems, such as modeling population growth, predicting stock prices, or determining the cost of a product. By understanding the range of a function, you can make more informed decisions and predictions.

Conclusion

In this article, we have answered some frequently asked questions related to finding the range of a product of functions. We have discussed the concept of product of functions, domain and range, and algebraic manipulation. We have also provided examples and explanations to help you understand the material.

Key Takeaways

The product of two functions is defined as $(u \cdot v)(x) = u(x) \cdot v(x)$ .
The domain of a function is the set of all possible input values for which the function is defined.
The range of a function is the set of all possible output values of the function.
A rational function has a range that includes all real numbers except for the values that make the numerator zero.

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Algebra, 2nd edition, Michael Artin

Additional Resources

Khan Academy: Functions and Graphs
MIT OpenCourseWare: Calculus
Wolfram Alpha: Function Operations