Select The Correct Answer.Carty Is Traveling To Visit Family. She Drives $x$ Miles On The First Day And $y$ Miles On The Second Day For A Total Of 800 Miles.What Are The Domain And Range Of This Relationship?A. Domain: $0

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Introduction

In mathematics, a relationship between two variables is often represented as an equation or a graph. The domain and range of a relationship are crucial concepts that help us understand the possible values of the variables involved. In this article, we will explore the domain and range of a relationship between the distances traveled by Carty on her first and second days of a trip.

The Problem

Carty is traveling to visit family. She drives $x$ miles on the first day and $y$ miles on the second day for a total of 800 miles. We need to find the domain and range of this relationship.

Defining the Domain and Range

The domain of a relationship is the set of all possible input values (in this case, the distances traveled on the first and second days). The range of a relationship is the set of all possible output values (in this case, the total distance traveled).

Finding the Domain

To find the domain, we need to consider the possible values of $x$ and $y$. Since Carty drives a non-negative distance on each day, we can assume that $x \geq 0$ and $y \geq 0$. Additionally, the total distance traveled is 800 miles, so we can write the equation:

x+y=800x + y = 800

We can solve this equation for $y$ to get:

y=800−xy = 800 - x

Since $y$ must be non-negative, we can set up the inequality:

800−x≥0800 - x \geq 0

Solving for $x$, we get:

x≤800x \leq 800

Therefore, the domain of the relationship is:

Domain: 0≤x≤800\text{Domain: } 0 \leq x \leq 800

Finding the Range

To find the range, we need to consider the possible values of the total distance traveled. Since the total distance is 800 miles, we can write:

0≤x+y≤8000 \leq x + y \leq 800

Substituting $y = 800 - x$, we get:

0≤x+(800−x)≤8000 \leq x + (800 - x) \leq 800

Simplifying, we get:

0≤800≤8000 \leq 800 \leq 800

This inequality is always true, so the range of the relationship is:

Range: 0≤x+y≤800\text{Range: } 0 \leq x + y \leq 800

Conclusion

In conclusion, the domain and range of the relationship between the distances traveled by Carty on her first and second days of a trip are:

  • Domain: $0 \leq x \leq 800$
  • Range: $0 \leq x + y \leq 800$

These results make sense, as Carty can travel any distance between 0 and 800 miles on each day, and the total distance traveled must be between 0 and 800 miles.

Example Use Cases

  1. Travel Planning: If Carty wants to plan a trip and knows that she will drive a certain distance on each day, she can use the domain and range to determine the possible total distance traveled.
  2. Budgeting: If Carty has a budget for her trip and knows that she will drive a certain distance on each day, she can use the domain and range to determine the possible total cost of the trip.

Tips and Variations

  1. Multiple Variables: If Carty has multiple variables, such as the distance traveled on each day and the number of days traveled, the domain and range can be more complex.
  2. Inequalities: If Carty has inequalities, such as $x + y \geq 800$, the domain and range can be more complex.
  3. Graphing: If Carty wants to graph the relationship between the distances traveled on each day, she can use the domain and range to determine the possible values of the variables.

Frequently Asked Questions

Q: What is the domain of a relationship?

A: The domain of a relationship is the set of all possible input values. In the context of Carty's trip, the domain is the set of all possible distances traveled on the first and second days.

Q: What is the range of a relationship?

A: The range of a relationship is the set of all possible output values. In the context of Carty's trip, the range is the set of all possible total distances traveled.

Q: How do I find the domain of a relationship?

A: To find the domain, you need to consider the possible values of the input variables. In the case of Carty's trip, we assumed that the distance traveled on each day is non-negative, so we set up the inequality $x \geq 0$ and $y \geq 0$. We then solved the equation $x + y = 800$ to get $y = 800 - x$ and set up the inequality $800 - x \geq 0$ to get $x \leq 800$.

Q: How do I find the range of a relationship?

A: To find the range, you need to consider the possible values of the output variable. In the case of Carty's trip, we set up the inequality $0 \leq x + y \leq 800$ and solved it to get $0 \leq x + y \leq 800$.

Q: What if I have multiple variables?

A: If you have multiple variables, the domain and range can be more complex. You need to consider the possible values of each variable and set up the corresponding inequalities.

Q: What if I have inequalities?

A: If you have inequalities, the domain and range can be more complex. You need to consider the possible values of each variable and set up the corresponding inequalities.

Q: How do I graph a relationship?

A: To graph a relationship, you need to plot the possible values of the input and output variables. In the case of Carty's trip, you can plot the possible values of $x$ and $y$ and the corresponding total distance traveled.

Q: What are some real-world applications of domain and range?

A: Domain and range have many real-world applications, including:

  • Travel planning: You can use domain and range to determine the possible total distance traveled on a trip.
  • Budgeting: You can use domain and range to determine the possible total cost of a trip.
  • Science: You can use domain and range to determine the possible values of variables in scientific experiments.
  • Engineering: You can use domain and range to determine the possible values of variables in engineering designs.

Q: What are some common mistakes to avoid when working with domain and range?

A: Some common mistakes to avoid when working with domain and range include:

  • Not considering the possible values of each variable.
  • Not setting up the correct inequalities.
  • Not solving the inequalities correctly.
  • Not considering the possible values of the output variable.

Q: How can I practice working with domain and range?

A: You can practice working with domain and range by:

  • Solving problems that involve domain and range.
  • Graphing relationships and determining the domain and range.
  • Using real-world examples to practice working with domain and range.

By understanding the domain and range of a relationship, you can make informed decisions and solve problems more effectively.