Finding the Domain of the Daily Production Cost Function

Oct 13, 2025 by miodeya.com 107 views

Introduction

In the world of economics and business, understanding the daily production cost is crucial for making informed decisions. The daily production cost, denoted as ${$ C $}$, for ${$ x $}$ units is modeled by the equation: ${ C = 200 - 7x + 0.345x^2 }$ . In this article, we will delve into the world of quadratic equations and explore how to find the domain and range of the daily production cost function.

What is the Domain of a Function?

The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of ${$ x $}$ for which the function ${$ C $}$ is defined. To find the domain of a function, we need to consider the restrictions on the input values.

Finding the Domain of the Daily Production Cost Function

The daily production cost function is a quadratic equation, which means it is a polynomial of degree two. The general form of a quadratic equation is: ${ ax^2 + bx + c }$ . In this case, the equation is: ${ C = 200 - 7x + 0.345x^2 }$ . Since the equation is a polynomial, it is defined for all real numbers. Therefore, the domain of the daily production cost function is all real numbers, denoted as ${ (-\infty, \infty) }$ .

What is the Range of a Function?

The range of a function is the set of all possible output values for which the function is defined. In other words, it is the set of all possible values of ${$ C $}$ for which the function is defined. To find the range of a function, we need to consider the minimum and maximum values of the function.

Finding the Range of the Daily Production Cost Function

To find the range of the daily production cost function, we need to find the minimum and maximum values of the function. Since the function is a quadratic equation, it has a parabolic shape. The minimum value of the function occurs at the vertex of the parabola, which can be found using the formula: ${ x = -\frac{b}{2a} }$ . In this case, the equation is: ${ C = 200 - 7x + 0.345x^2 }$ . The coefficient of the squared term is ${ a = 0.345 }$ , and the coefficient of the linear term is ${ b = -7 }$ . Plugging these values into the formula, we get: ${ x = -\frac{-7}{2(0.345)} }$ . Simplifying, we get: ${ x = \frac{7}{0.69} }$ . Therefore, the minimum value of the function occurs at ${ x = 10.14 }$ .

Finding the Minimum Value of the Daily Production Cost Function

To find the minimum value of the function, we plug the value of ${ x }$ into the equation: ${ C = 200 - 7x + 0.345x^2 }$ . Plugging in ${ x = 10.14 }$ , we get: ${ C = 200 - 7(10.14) + 0.345(10.14)^2 }$ . Simplifying, we get: ${ C = 200 - 71.98 + 36.31 }$ . Therefore, the minimum value of the function is: ${ C = 164.33 }$ .

Finding the Maximum Value of the Daily Production Cost Function

Since the function is a quadratic equation, it has a parabolic shape. The maximum value of the function occurs at the endpoints of the domain. Since the domain is all real numbers, the endpoints are ${ x = -\infty }$ and ${ x = \infty }$ . However, since the function is a quadratic equation, it approaches the horizontal asymptote as ${ x }$ approaches ${ \pm \infty }$ . The horizontal asymptote is the value that the function approaches as ${ x }$ approaches ${ \pm \infty }$ . To find the horizontal asymptote, we look at the leading term of the equation, which is ${ 0.345x^2 }$ . As ${ x }$ approaches ${ \pm \infty }$ , the term ${ 0.345x^2 }$ dominates the equation. Therefore, the horizontal asymptote is: ${ y = 0 }$ .

Conclusion

In conclusion, the domain of the daily production cost function is all real numbers, denoted as ${ (-\infty, \infty) }$ . The range of the function is the set of all possible output values for which the function is defined. To find the range of the function, we need to find the minimum and maximum values of the function. The minimum value of the function occurs at the vertex of the parabola, which can be found using the formula: ${ x = -\frac{b}{2a} }$ . The maximum value of the function occurs at the endpoints of the domain, which are ${ x = -\infty }$ and ${ x = \infty }$ . However, since the function is a quadratic equation, it approaches the horizontal asymptote as ${ x }$ approaches ${ \pm \infty }$ . The horizontal asymptote is the value that the function approaches as ${ x }$ approaches ${ \pm \infty }$ . Therefore, the range of the function is: ${ (-\infty, 200) }$ .

References

[1] "Quadratic Equations" by Math Open Reference
[2] "Domain and Range" by Khan Academy
[3] "Quadratic Functions" by Purplemath

Additional Resources

[1] "Quadratic Equations" by Wolfram MathWorld
[2] "Domain and Range" by IXL
[3] "Quadratic Functions" by Mathway
The Daily Production Cost: A Q&A Guide =====================================

Introduction

In our previous article, we explored the daily production cost function and how to find its domain and range. In this article, we will answer some frequently asked questions about the daily production cost function.

Q: What is the daily production cost function?

A: The daily production cost function is a quadratic equation that models the daily production cost for a given number of units. The equation is: ${ C = 200 - 7x + 0.345x^2 }$ .

Q: What is the domain of the daily production cost function?

A: The domain of the daily production cost function is all real numbers, denoted as ${ (-\infty, \infty) }$ .

Q: What is the range of the daily production cost function?

A: The range of the daily production cost function is the set of all possible output values for which the function is defined. To find the range of the function, we need to find the minimum and maximum values of the function.

Q: How do I find the minimum value of the daily production cost function?

A: To find the minimum value of the function, we plug the value of ${ x }$ into the equation: ${ C = 200 - 7x + 0.345x^2 }$ . The minimum value of the function occurs at the vertex of the parabola, which can be found using the formula: ${ x = -\frac{b}{2a} }$ .

Q: How do I find the maximum value of the daily production cost function?

A: Since the function is a quadratic equation, it has a parabolic shape. The maximum value of the function occurs at the endpoints of the domain. However, since the function is a quadratic equation, it approaches the horizontal asymptote as ${ x }$ approaches ${ \pm \infty }$ . The horizontal asymptote is the value that the function approaches as ${ x }$ approaches ${ \pm \infty }$ .

Q: What is the horizontal asymptote of the daily production cost function?

A: The horizontal asymptote of the daily production cost function is the value that the function approaches as ${ x }$ approaches ${ \pm \infty }$ . In this case, the horizontal asymptote is: ${ y = 0 }$ .

Q: How do I use the daily production cost function in real-world applications?

A: The daily production cost function can be used in a variety of real-world applications, such as:

Manufacturing: The daily production cost function can be used to determine the cost of producing a given number of units.
Supply Chain Management: The daily production cost function can be used to determine the cost of transporting goods from one location to another.
Economics: The daily production cost function can be used to determine the cost of producing a given number of units in a given market.

Q: What are some common mistakes to avoid when working with the daily production cost function?

A: Some common mistakes to avoid when working with the daily production cost function include:

Not considering the domain and range of the function: The domain and range of the function are critical in determining the validity of the function.
Not considering the minimum and maximum values of the function: The minimum and maximum values of the function are critical in determining the cost of producing a given number of units.
Not considering the horizontal asymptote of the function: The horizontal asymptote of the function is critical in determining the cost of producing a given number of units as ${ x }$ approaches ${ \pm \infty }$ .

Conclusion

In conclusion, the daily production cost function is a quadratic equation that models the daily production cost for a given number of units. The domain of the function is all real numbers, and the range of the function is the set of all possible output values for which the function is defined. To find the range of the function, we need to find the minimum and maximum values of the function. The minimum value of the function occurs at the vertex of the parabola, and the maximum value of the function occurs at the endpoints of the domain. The horizontal asymptote of the function is the value that the function approaches as ${ x }$ approaches ${ \pm \infty }$ . By understanding the daily production cost function, we can use it in a variety of real-world applications, such as manufacturing, supply chain management, and economics.

References

[1] "Quadratic Equations" by Math Open Reference
[2] "Domain and Range" by Khan Academy
[3] "Quadratic Functions" by Purplemath

Additional Resources

[1] "Quadratic Equations" by Wolfram MathWorld
[2] "Domain and Range" by IXL
[3] "Quadratic Functions" by Mathway