A Rectangle Has An Area Of $24 , \text{cm}^2$. Function $f$ Gives The Length Of The Rectangle, In Centimeters, When The Width Is $\omega , \text{cm}$.Determine If Each Value, In Centimeters, Is A Possible Input Of

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Introduction

When dealing with geometric shapes, understanding the relationship between their dimensions is crucial. In this case, we are given a rectangle with an area of $24 , \text{cm}^2$ and a function $f$ that gives the length of the rectangle in centimeters when the width is $\omega , \text{cm}$. Our goal is to determine if each value, in centimeters, is a possible input of the function $f$.

Understanding the Function

The function $f$ is defined as the length of the rectangle when the width is $\omega , \text{cm}$. This means that for any given width $\omega$, the function $f$ will return the corresponding length of the rectangle. To find the length of the rectangle, we can use the formula for the area of a rectangle, which is given by:

A=l×wA = l \times w

where $A$ is the area, $l$ is the length, and $w$ is the width.

Finding the Length of the Rectangle

Given that the area of the rectangle is $24 , \text{cm}^2$, we can use the formula for the area to find the length of the rectangle. We can rearrange the formula to solve for the length:

l=Awl = \frac{A}{w}

Substituting the given area of $24 , \text{cm}^2$, we get:

l=24wl = \frac{24}{w}

This means that the function $f$ can be defined as:

f(ω)=24ωf(\omega) = \frac{24}{\omega}

Determining Possible Inputs

Now that we have defined the function $f$, we need to determine if each value, in centimeters, is a possible input of the function. In other words, we need to find the values of $\omega$ for which the function $f$ is defined.

Possible Inputs

To find the possible inputs of the function $f$, we need to consider the domain of the function. The domain of a function is the set of all possible input values for which the function is defined.

In this case, the function $f$ is defined as:

f(ω)=24ωf(\omega) = \frac{24}{\omega}

The function is defined for all values of $\omega$ except for $\omega = 0$, since division by zero is undefined.

Conclusion

In conclusion, the possible inputs of the function $f$ are all values of $\omega$ except for $\omega = 0$. This means that the function $f$ is defined for all positive and negative values of $\omega$, except for $\omega = 0$.

Example

Let's consider an example to illustrate this concept. Suppose we want to find the length of the rectangle when the width is $4 , \text{cm}$. We can use the function $f$ to find the length:

f(4)=244=6f(4) = \frac{24}{4} = 6

This means that the length of the rectangle is $6 , \text{cm}$ when the width is $4 , \text{cm}$.

Graphing the Function

To visualize the function $f$, we can graph it on a coordinate plane. The graph of the function $f$ will be a hyperbola, since the function is defined as the ratio of two variables.

Graph

Here is a graph of the function $f$:

import numpy as np
import matplotlib.pyplot as plt

def f(x):
return 24 / x



x = np.linspace(0.1, 10, 100)



y = f(x)



plt.plot(x, y)
plt.xlabel('Width (cm)')
plt.ylabel('Length (cm)')
plt.title('Graph of the Function f')
plt.grid(True)
plt.show()

This graph shows the relationship between the width and the length of the rectangle. The graph is a hyperbola, since the function is defined as the ratio of two variables.

Final Thoughts

In conclusion, the function $f$ is defined as the length of the rectangle when the width is $\omega , \text{cm}$. The possible inputs of the function $f$ are all values of $\omega$ except for $\omega = 0$. The graph of the function $f$ is a hyperbola, which shows the relationship between the width and the length of the rectangle.

References

  • [1] "Functions and Graphs". Khan Academy.
  • [2] "Geometry". Math Open Reference.
  • [3] "Hyperbolas". Math Is Fun.

Discussion

This problem is a classic example of a function that is defined as the ratio of two variables. The function $f$ is defined as the length of the rectangle when the width is $\omega , \text{cm}$. The possible inputs of the function $f$ are all values of $\omega$ except for $\omega = 0$. The graph of the function $f$ is a hyperbola, which shows the relationship between the width and the length of the rectangle.

Related Problems

  • [1] "Find the length of a rectangle when the width is $5 , \text{cm}$".
  • [2] "Graph the function $f(x) = \frac{1}{x}$".
  • [3] "Find the domain of the function $f(x) = \frac{1}{x^2}$".

Tags

  • Functions
  • Graphs
  • Hyperbolas
  • Geometry
  • Math
  • Problem Solving

Introduction

In our previous article, we explored the function $f$ that gives the length of a rectangle in centimeters when the width is $\omega , \text{cm}$. We determined that the possible inputs of the function $f$ are all values of $\omega$ except for $\omega = 0$. In this article, we will answer some frequently asked questions about the function and its possible inputs.

Q: What is the function $f$ defined as?

A: The function $f$ is defined as the length of the rectangle when the width is $\omega , \text{cm}$. It can be expressed as:

f(ω)=24ωf(\omega) = \frac{24}{\omega}

Q: What are the possible inputs of the function $f$?

A: The possible inputs of the function $f$ are all values of $\omega$ except for $\omega = 0$. This means that the function $f$ is defined for all positive and negative values of $\omega$, except for $\omega = 0$.

Q: Why is $\omega = 0$ not a possible input of the function $f$?

A: $\omega = 0$ is not a possible input of the function $f$ because division by zero is undefined. When $\omega = 0$, the function $f$ would be undefined, which is not a valid input.

Q: Can I use the function $f$ to find the length of a rectangle when the width is $0 , \text{cm}$?

A: No, you cannot use the function $f$ to find the length of a rectangle when the width is $0 , \text{cm}$. As we discussed earlier, $\omega = 0$ is not a possible input of the function $f$.

Q: How can I graph the function $f$?

A: You can graph the function $f$ by using a coordinate plane and plotting the points that satisfy the equation $f(\omega) = \frac{24}{\omega}$. The graph of the function $f$ will be a hyperbola.

Q: What is the relationship between the width and the length of the rectangle?

A: The relationship between the width and the length of the rectangle is given by the function $f$. The function $f$ shows that the length of the rectangle is inversely proportional to the width.

Q: Can I use the function $f$ to find the width of a rectangle when the length is given?

A: No, the function $f$ is defined as the length of the rectangle when the width is given. It is not possible to use the function $f$ to find the width of a rectangle when the length is given.

Q: What are some real-world applications of the function $f$?

A: The function $f$ has many real-world applications, such as finding the length of a rectangle when the width is given, or finding the width of a rectangle when the length is given. It can also be used in engineering and architecture to design buildings and other structures.

Q: Can I use the function $f$ to solve other problems?

A: Yes, the function $f$ can be used to solve other problems that involve finding the length or width of a rectangle. It can also be used to solve problems that involve inverse proportions.

Conclusion

In this article, we answered some frequently asked questions about the function $f$ and its possible inputs. We discussed the definition of the function $f$, the possible inputs of the function $f$, and the relationship between the width and the length of the rectangle. We also explored some real-world applications of the function $f$ and discussed how it can be used to solve other problems.

Final Thoughts

The function $f$ is a powerful tool that can be used to solve many problems that involve finding the length or width of a rectangle. It is an important concept in mathematics and has many real-world applications. By understanding the function $f$ and its possible inputs, you can solve a wide range of problems and make informed decisions in your personal and professional life.

References

  • [1] "Functions and Graphs". Khan Academy.
  • [2] "Geometry". Math Open Reference.
  • [3] "Hyperbolas". Math Is Fun.

Discussion

This article is a continuation of our previous article on the function $f$. We explored the function $f$ and its possible inputs, and answered some frequently asked questions about the function. We also discussed some real-world applications of the function $f$ and explored how it can be used to solve other problems.

Related Problems

  • [1] "Find the length of a rectangle when the width is $5 , \text{cm}$".
  • [2] "Graph the function $f(x) = \frac{1}{x}$".
  • [3] "Find the domain of the function $f(x) = \frac{1}{x^2}$".

Tags

  • Functions
  • Graphs
  • Hyperbolas
  • Geometry
  • Math
  • Problem Solving