The Product of the Functions

Oct 13, 2025 by windailys.com 30 views

Understanding the Problem: A Comprehensive Analysis of the Range of a Product of Functions

In this article, we will delve into the world of mathematics, specifically focusing on the concept of functions and their products. We will explore the given problem, which involves finding the range of the product of two functions, $u(x)$ and $v(x)$ . Our goal is to provide a clear and concise explanation of the problem, along with a step-by-step solution, to help readers understand the underlying concepts.

The Functions $u(x)$ and $v(x)$

The two functions given in the problem are:

u(x)=-2x^2+3

v(x)=\frac{1}{x}

These functions are defined for all real values of $x$ , except for $x=0$ , since the function $v(x)$ is not defined at $x=0$ .

The Product of the Functions

To find the range of the product of the two functions, we need to first find the product of the functions, denoted as $(u \cdot v)(x)$ . This can be done by multiplying the two functions together:

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

Simplifying the expression, we get:

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

Finding the Range of the Product

To find the range of the product of the two functions, we need to consider the possible values that the product can take. Since the product is a continuous function, we can use the concept of limits to find the range.

Let's consider the behavior of the product as $x$ approaches positive and negative infinity:

As $x$ approaches positive infinity, the term $-2x$ dominates the expression, and the product approaches negative infinity.
As $x$ approaches negative infinity, the term $-2x$ also dominates the expression, and the product approaches positive infinity.

Now, let's consider the behavior of the product as $x$ approaches 0 from the right and left:

As $x$ approaches 0 from the right, the term $\frac{3}{x}$ dominates the expression, and the product approaches positive infinity.
As $x$ approaches 0 from the left, the term $\frac{3}{x}$ also dominates the expression, and the product approaches negative infinity.

From these observations, we can conclude that the product of the two functions can take on any real value, except for 0.

Conclusion

In conclusion, the range of the product of the two functions $u(x)$ and $v(x)$ is all real numbers, except for 0. This can be represented mathematically as:

(-\infty, 0) \cup (0, \infty)

Therefore, the correct answer is:

D. $(-\infty, +\infty)$

Key Takeaways

The product of two functions can be found by multiplying the two functions together.
The range of a product of functions can be found by considering the possible values that the product can take.
The concept of limits can be used to find the range of a product of functions.
The range of a product of functions can be represented mathematically using interval notation.

Final Thoughts

In this article, we have explored the concept of finding the range of a product of functions. We have seen how to find the product of two functions and how to use the concept of limits to find the range. We have also seen how to represent the range mathematically using interval notation. We hope that this article has provided a clear and concise explanation of the problem and has helped readers understand the underlying concepts.
Q&A: Understanding the Range of a Product of Functions

In our previous article, we explored the concept of finding the range of a product of functions. We discussed how to find the product of two functions and how to use the concept of limits to find the range. In this article, we will answer some frequently asked questions related to the topic.

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

A: The range of a product of functions is the set of all possible values that the product can take. In other words, it is the set of all possible outputs of the product function.

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

A: To find the range of a product of functions, you need to consider the possible values that the product can take. You can do this by analyzing the behavior of the product as the input variable approaches positive and negative infinity, as well as the behavior of the product as the input variable approaches specific values.

Q: What is the difference between the range and the domain 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 the range of a product of functions be infinite?

A: Yes, the range of a product of functions can be infinite. This is because the product of two functions can take on any real value, including positive and negative infinity.

Q: How do I represent the range of a product of functions mathematically?

A: You can represent the range of a product of functions mathematically using interval notation. For example, if the range of a product of functions is all real numbers except for 0, you can represent it as:

(-\infty, 0) \cup (0, \infty)

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

A: The range of a product of functions is significant because it helps us understand the behavior of the product function. By knowing the range of a product of functions, we can determine the possible outputs of the function and make predictions about its behavior.

Q: Can the range of a product of functions be a single value?

A: Yes, the range of a product of functions can be a single value. This is because the product of two functions can take on a specific value, depending on the input values.

Q: How do I find the range of a product of functions with multiple variables?

A: To find the range of a product of functions with multiple variables, you need to consider the possible values that the product can take for each variable. You can do this by analyzing the behavior of the product as each variable approaches positive and negative infinity, as well as the behavior of the product as each variable approaches specific values.

Q: What is the relationship between the range of a product of functions and the range of each individual function?

A: The range of a product of functions is related to the range of each individual function. Specifically, the range of the product of two functions is the set of all possible products of the ranges of the two individual functions.

Q: Can the range of a product of functions be a subset of the range of one of the individual functions?

A: Yes, the range of a product of functions can be a subset of the range of one of the individual functions. This is because the product of two functions can take on values that are not present in the range of one of the individual functions.

Q: How do I determine the range of a product of functions with trigonometric functions?

A: To determine the range of a product of functions with trigonometric functions, you need to consider the possible values that the product can take. You can do this by analyzing the behavior of the product as the input variable approaches positive and negative infinity, as well as the behavior of the product as the input variable approaches specific values.

Q: Can the range of a product of functions be a periodic function?

A: Yes, the range of a product of functions can be a periodic function. This is because the product of two functions can take on values that are periodic, depending on the input values.

Q: How do I find the range of a product of functions with exponential functions?

A: To find the range of a product of functions with exponential functions, you need to consider the possible values that the product can take. You can do this by analyzing the behavior of the product as the input variable approaches positive and negative infinity, as well as the behavior of the product as the input variable approaches specific values.

Q: Can the range of a product of functions be a constant function?

A: Yes, the range of a product of functions can be a constant function. This is because the product of two functions can take on a specific value, depending on the input values.

Q: How do I determine the range of a product of functions with rational functions?

A: To determine the range of a product of functions with rational functions, you need to consider the possible values that the product can take. You can do this by analyzing the behavior of the product as the input variable approaches positive and negative infinity, as well as the behavior of the product as the input variable approaches specific values.

Q: Can the range of a product of functions be a function with a hole?

A: Yes, the range of a product of functions can be a function with a hole. This is because the product of two functions can take on values that are not present in the range of one of the individual functions.

Q: How do I find the range of a product of functions with a piecewise function?

A: To find the range of a product of functions with a piecewise function, you need to consider the possible values that the product can take for each piece of the function. You can do this by analyzing the behavior of the product as the input variable approaches positive and negative infinity, as well as the behavior of the product as the input variable approaches specific values.

Q: Can the range of a product of functions be a function with a vertical asymptote?

A: Yes, the range of a product of functions can be a function with a vertical asymptote. This is because the product of two functions can take on values that are not present in the range of one of the individual functions.

Q: How do I determine the range of a product of functions with a function that is not defined at a specific point?

A: To determine the range of a product of functions with a function that is not defined at a specific point, you need to consider the possible values that the product can take. You can do this by analyzing the behavior of the product as the input variable approaches positive and negative infinity, as well as the behavior of the product as the input variable approaches specific values.

Q: Can the range of a product of functions be a function with a horizontal asymptote?

A: Yes, the range of a product of functions can be a function with a horizontal asymptote. This is because the product of two functions can take on values that are not present in the range of one of the individual functions.

Q: How do I find the range of a product of functions with a function that is periodic?

A: To find the range of a product of functions with a function that is periodic, you need to consider the possible values that the product can take. You can do this by analyzing the behavior of the product as the input variable approaches positive and negative infinity, as well as the behavior of the product as the input variable approaches specific values.

Q: Can the range of a product of functions be a function with a discontinuity?

A: Yes, the range of a product of functions can be a function with a discontinuity. This is because the product of two functions can take on values that are not present in the range of one of the individual functions.

Q: How do I determine the range of a product of functions with a function that is not continuous?

A: To determine the range of a product of functions with a function that is not continuous, you need to consider the possible values that the product can take. You can do this by analyzing the behavior of the product as the input variable approaches positive and negative infinity, as well as the behavior of the product as the input variable approaches specific values.

Q: Can the range of a product of functions be a function with a limit that does not exist?

A: Yes, the range of a product of functions can be a function with a limit that does not exist. This is because the product of two functions can take on values that are not present in the range of one of the individual functions.

Q: How do I find the range of a product of functions with a function that has a limit that does not exist?

A: To find the range of a product of functions with a function that has a limit that does not exist, you need to consider the possible values that the product can take. You can do this by analyzing the behavior of the product as the input variable approaches positive and negative infinity, as well as the behavior of the product as the input variable approaches specific values.

Q: Can the range of a product of functions be a function with a discontinuity at a specific point?

A: Yes, the range of a product of functions can be a function with a discontinuity at a specific point. This is because the product of two functions can take on values that are not present in the range of one of the individual functions.

Q: How do I determine the range of a product of functions with a function that has a discontinuity at a specific point?

A: To determine the range of a product of functions with a function that has a discontinuity at