The Number Of Songs That Can Be Purchased For $\$1.50$ Each Based On The Amount Of The Gift Card Is Represented By The Function $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$. Which Graph Represents This Situation?

Feb 26, 2025 by ADMIN 215 views

$The Number of Songs That Can Be Purchased for $\$1.50$ Each Based on the Amount of the Gift Card$

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Introduction

The problem presents a scenario where the number of songs that can be purchased for $\$1.50$ each is represented by a function $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ . This function takes the total amount of the gift card as input and returns the number of songs that can be purchased. In this article, we will explore the graph that represents this situation.

Understanding the Function

The function $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ is a floor function, which means it returns the greatest integer less than or equal to the result of the division. This function can be broken down into two parts:

The division $\frac{x}{1.50}$ represents the number of songs that can be purchased if there were no floor function.
The floor function $\left\lfloor\cdot\right\rfloor$ rounds down the result of the division to the nearest integer.

Graphing the Function

To graph the function $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ , we need to consider the following:

For $x<1.50$ , the function is not defined, as the division $\frac{x}{1.50}$ would result in a non-integer value.
For $1.50\leq x<3$ , the function evaluates to 1, as the floor function rounds down the result of the division to the nearest integer.
For $3\leq x<4.50$ , the function evaluates to 2, as the floor function rounds down the result of the division to the nearest integer.
This pattern continues, with the function evaluating to $n$ for $1.50n\leq x<1.50(n+1)$ .

Graph Representation

The graph of the function $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ is a step function, with each step representing the number of songs that can be purchased for a given amount of the gift card. The graph consists of a series of horizontal line segments, with each segment representing a range of values for $x$ .

Key Features of the Graph

The graph of the function $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ has the following key features:

Step Function: The graph is a step function, with each step representing the number of songs that can be purchased for a given amount of the gift card.
Horizontal Line Segments: The graph consists of a series of horizontal line segments, with each segment representing a range of values for $x$ .
Discontinuities: The graph has discontinuities at $x=1.50, 3, 4.50, ...$ , as the function evaluates to a different value at each of these points.

Conclusion

In conclusion, the graph that represents the situation is a step function with horizontal line segments, representing the number of songs that can be purchased for a given amount of the gift card. The graph has discontinuities at $x=1.50, 3, 4.50, ...$ , as the function evaluates to a different value at each of these points.

References

[1] Floor Function. (n.d.). In Encyclopedia of Mathematics. Retrieved from https://encyclopediaofmath.org/wiki/Floor_function
[2] Step Function. (n.d.). In Encyclopedia of Mathematics. Retrieved from https://encyclopediaofmath.org/wiki/Step_function

Q: What is the purpose of the floor function in the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

A: The floor function is used to round down the result of the division $\frac{x}{1.50}$ to the nearest integer. This is necessary because the number of songs that can be purchased must be an integer.

Q: What is the significance of the horizontal line segments in the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

A: The horizontal line segments in the graph represent the number of songs that can be purchased for a given amount of the gift card. Each segment corresponds to a range of values for $x$ .

Q: Where are the discontinuities in the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

A: The discontinuities in the graph occur at $x=1.50, 3, 4.50, ...$ , as the function evaluates to a different value at each of these points.

Q: How does the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ relate to the concept of step functions?

A: The graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ is an example of a step function, which is a mathematical function that consists of a series of horizontal line segments.

Q: What is the relationship between the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ and the concept of floor functions?

A: The graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ is an example of a floor function, which is a mathematical function that returns the greatest integer less than or equal to a given number.

Q: How does the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ relate to real-world applications?

A: The graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ can be used to model real-world situations where the number of items that can be purchased is limited by a fixed cost.

Q: What are some common mistakes to avoid when graphing the function $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

A: Some common mistakes to avoid when graphing the function $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ include:

Failing to account for the floor function
Not recognizing the step function pattern
Misinterpreting the discontinuities

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to solve real-world problems?

A: The graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ can be used to solve real-world problems by modeling the number of items that can be purchased given a fixed cost.

Q: What are some additional resources that I can use to learn more about the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

A: Some additional resources that you can use to learn more about the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ include:

Online tutorials and videos
Math textbooks and workbooks
Online forums and communities

Q: How can I apply the concepts learned from the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to other areas of mathematics?

A: The concepts learned from the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ can be applied to other areas of mathematics, such as:

Algebra
Calculus
Statistics

Q: What are some common applications of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in real-world scenarios?

A: Some common applications of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in real-world scenarios include:

Modeling the number of items that can be purchased given a fixed cost
Analyzing the relationship between the number of items and the cost
Making predictions about the number of items that can be purchased given a certain cost

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to make informed decisions in real-world scenarios?

A: The graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ can be used to make informed decisions in real-world scenarios by:

Analyzing the relationship between the number of items and the cost
Making predictions about the number of items that can be purchased given a certain cost
Identifying the optimal number of items to purchase given a fixed cost

Q: What are some common challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

A: Some common challenges that you may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ include:

Difficulty understanding the floor function
Difficulty recognizing the step function pattern
Difficulty interpreting the discontinuities

Q: How can I overcome the challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

A: You can overcome the challenges that you may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ by:

Seeking help from a teacher or tutor
Practicing with examples and exercises
Using online resources and tutorials

Q: What are some additional tips that I can use to improve my understanding of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

A: Some additional tips that you can use to improve your understanding of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ include:

Practice, practice, practice!
Use online resources and tutorials
Seek help from a teacher or tutor

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to solve problems in other areas of mathematics?

A: The graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ can be used to solve problems in other areas of mathematics, such as:

Algebra
Calculus
Statistics

Q: What are some common applications of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in other areas of mathematics?

A: Some common applications of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in other areas of mathematics include:

Modeling the number of items that can be purchased given a fixed cost
Analyzing the relationship between the number of items and the cost
Making predictions about the number of items that can be purchased given a certain cost

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to make informed decisions in other areas of mathematics?

A: The graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ can be used to make informed decisions in other areas of mathematics by:

Analyzing the relationship between the number of items and the cost
Making predictions about the number of items that can be purchased given a certain cost
Identifying the optimal number of items to purchase given a fixed cost

Q: What are some common challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in other areas of mathematics?

A: Some common challenges that you may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in other areas of mathematics include:

Difficulty understanding the floor function
Difficulty recognizing the step function pattern
Difficulty interpreting the discontinuities

Q: How can I overcome the challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in other areas of mathematics?

A: You can overcome the challenges

The Number Of Songs That Can Be Purchased For $\$1.50$ Each Based On The Amount Of The Gift Card Is Represented By The Function $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$. Which Graph Represents This Situation?

Introduction

Understanding the Function

Graphing the Function

Graph Representation

Key Features of the Graph

Conclusion

References

Further Reading

Related Topics

Tags

Q: What is the purpose of the floor function in the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: What is the significance of the horizontal line segments in the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: Where are the discontinuities in the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: How does the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ relate to the concept of step functions?

Q: What is the relationship between the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ and the concept of floor functions?

Q: How does the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ relate to real-world applications?

Q: What are some common mistakes to avoid when graphing the function $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to solve real-world problems?

Q: What are some additional resources that I can use to learn more about the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: How can I apply the concepts learned from the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to other areas of mathematics?

Q: What are some common applications of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in real-world scenarios?

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to make informed decisions in real-world scenarios?

Q: What are some common challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: How can I overcome the challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: What are some additional tips that I can use to improve my understanding of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to solve problems in other areas of mathematics?

Q: What are some common applications of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in other areas of mathematics?

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to make informed decisions in other areas of mathematics?

Q: What are some common challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in other areas of mathematics?

Q: How can I overcome the challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in other areas of mathematics?

Introduction

Understanding the Function

Graphing the Function

Graph Representation

Key Features of the Graph

Conclusion

References

Further Reading

Related Topics

Tags

Q: What is the purpose of the floor function in the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋?

Q: What is the significance of the horizontal line segments in the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋?

Q: Where are the discontinuities in the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋?

Q: How does the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ relate to the concept of step functions?

Q: What is the relationship between the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ and the concept of floor functions?

Q: How does the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ relate to real-world applications?

Q: What are some common mistakes to avoid when graphing the function f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋?

Q: How can I use the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ to solve real-world problems?

Q: What are some additional resources that I can use to learn more about the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋?

Q: How can I apply the concepts learned from the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ to other areas of mathematics?

Q: What are some common applications of the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ in real-world scenarios?

Q: How can I use the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ to make informed decisions in real-world scenarios?

Q: What are some common challenges that I may face when working with the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋?

Q: How can I overcome the challenges that I may face when working with the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋?

Q: What are some additional tips that I can use to improve my understanding of the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋?

Q: How can I use the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ to solve problems in other areas of mathematics?

Q: What are some common applications of the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ in other areas of mathematics?

Q: How can I use the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ to make informed decisions in other areas of mathematics?

Q: What are some common challenges that I may face when working with the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ in other areas of mathematics?

Q: How can I overcome the challenges that I may face when working with the graph of f(x)=⌊x1.50⌋f(x)=\left\lfloor\frac{x}{1.50}\right\rfloorf(x)=⌊1.50x​⌋ in other areas of mathematics?

Q: What is the purpose of the floor function in the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: What is the significance of the horizontal line segments in the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: Where are the discontinuities in the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: How does the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ relate to the concept of step functions?

Q: What is the relationship between the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ and the concept of floor functions?

Q: How does the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ relate to real-world applications?

Q: What are some common mistakes to avoid when graphing the function $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to solve real-world problems?

Q: What are some additional resources that I can use to learn more about the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: How can I apply the concepts learned from the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to other areas of mathematics?

Q: What are some common applications of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in real-world scenarios?

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to make informed decisions in real-world scenarios?

Q: What are some common challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: How can I overcome the challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: What are some additional tips that I can use to improve my understanding of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ ?

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to solve problems in other areas of mathematics?

Q: What are some common applications of the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in other areas of mathematics?

Q: How can I use the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ to make informed decisions in other areas of mathematics?

Q: What are some common challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in other areas of mathematics?

Q: How can I overcome the challenges that I may face when working with the graph of $f(x)=\left\lfloor\frac{x}{1.50}\right\rfloor$ in other areas of mathematics?