The Function F ( A ) = 65 A F(a) = 65a F ( A ) = 65 A Shows The Distance You Are From Home On A Car Trip, Where A A A Is The Number Of Hours You Have Traveled. What Is The Domain Of This Function?A. A ≥ 0 A \geq 0 A ≥ 0 , Including Decimals To The Tenths Place B.

Introduction

When embarking on a car trip, it's essential to understand the relationship between the distance traveled and the time spent on the road. The function $f(a) = 65a$ represents this relationship, where $a$ is the number of hours traveled and $f(a)$ is the distance from home. In this article, we will delve into the domain of this function, exploring the possible values of $a$ that make the function valid.

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's the range of values that $a$ can take on, making the function $f(a) = 65a$ valid. To determine the domain, we need to consider the constraints on the input values, such as the presence of any restrictions or limitations.

Analyzing the Function

The function $f(a) = 65a$ is a linear function, meaning it has a constant slope of 65. This indicates that the distance traveled increases at a constant rate of 65 units per hour. However, this doesn't provide any information about the domain of the function.

Restrictions on Input Values

To determine the domain of the function, we need to consider any restrictions on the input values. In this case, the function is defined for all real numbers, but we need to consider the context of the problem. Since the function represents the distance traveled, it's reasonable to assume that the input values should be non-negative.

Why Non-Negative Input Values?

The reason for considering non-negative input values is that the distance traveled cannot be negative. If $a$ is negative, it would imply that the car has traveled a negative distance, which is physically impossible. Therefore, we can conclude that the domain of the function is restricted to non-negative values of $a$.

Domain of the Function

Based on the analysis above, we can conclude that the domain of the function $f(a) = 65a$ is:

• $a \geq 0$

This means that the function is defined for all non-negative values of $a$, including decimals to the tenths place.

Conclusion

In conclusion, the domain of the function $f(a) = 65a$ is $a \geq 0$, including decimals to the tenths place. This represents the set of all possible input values for which the function is defined, taking into account the context of the problem and the restrictions on the input values.

Final Answer

The final answer is:

• $a \geq 0$

This represents the domain of the function $f(a) = 65a$, which is the set of all possible input values for which the function is defined.

References

• [1] "Functions and Graphs" by Michael Sullivan
• [2] "Calculus" by James Stewart

Additional Resources

• Khan Academy: Functions and Graphs
• MIT OpenCourseWare: Calculus

Related Topics

• Domain and Range of a Function
• Linear Functions
• Graphing Functions

FAQs

• Q: What is the domain of the function $f(a) = 65a$? A: The domain of the function $f(a) = 65a$ is $a \geq 0$, including decimals to the tenths place.
• Q: Why is the domain restricted to non-negative values of $a$? A: The domain is restricted to non-negative values of $a$ because the distance traveled cannot be negative.
  Frequently Asked Questions (FAQs) About the Domain of a Function ====================================================================

Introduction

In our previous article, we explored the domain of the function $f(a) = 65a$, which represents the distance traveled on a car trip. We concluded that the domain of the function is $a \geq 0$, including decimals to the tenths place. In this article, we will address some frequently asked questions (FAQs) about the domain of a function.

Q: What is 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. In other words, it's the range of values that $a$ can take on, making the function valid.

Q: Why is the domain important?

A: The domain is important because it determines the set of input values for which the function is defined. If the domain is not properly defined, the function may not be valid or may produce incorrect results.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to consider the constraints on the input values, such as the presence of any restrictions or limitations. You should also consider the context of the problem and the type of function being used.

Q: What are some common restrictions on the domain of a function?

A: Some common restrictions on the domain of a function include:

• Non-negative values: The function may only be defined for non-negative values of $a$.
• Positive values: The function may only be defined for positive values of $a$.
• Integer values: The function may only be defined for integer values of $a$.
• Rational values: The function may only be defined for rational values of $a$.

Q: How do I represent the domain of a function mathematically?

A: The domain of a function can be represented mathematically using interval notation. For example, the domain of the function $f(a) = 65a$ can be represented as $[0, \infty)$, which means that the function is defined for all non-negative values of $a$.

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values for which the function is defined, while the range of a function is the set of all possible output values. In other words, the domain is the input values, while the range is the output values.

Q: Can the domain of a function be empty?

A: Yes, the domain of a function can be empty. This occurs when the function is not defined for any input values.

Q: Can the domain of a function be infinite?

A: Yes, the domain of a function can be infinite. This occurs when the function is defined for all real numbers.

Q: How do I graph a function with a restricted domain?

A: To graph a function with a restricted domain, you need to identify the restrictions on the input values and graph the function accordingly. For example, if the function is only defined for non-negative values of $a$, you would graph the function only for non-negative values of $a$.

Conclusion

In conclusion, the domain of a function is an essential concept in mathematics that determines the set of input values for which the function is defined. By understanding the domain of a function, you can ensure that the function is valid and produces correct results. We hope that this article has provided you with a better understanding of the domain of a function and how to determine it.

Final Answer

The final answer is:

• The domain of a function is the set of all possible input values for which the function is defined.

References

• [1] "Functions and Graphs" by Michael Sullivan
• [2] "Calculus" by James Stewart

Additional Resources

• Khan Academy: Functions and Graphs
• MIT OpenCourseWare: Calculus

Related Topics

• Domain and Range of a Function
• Linear Functions
• Graphing Functions

FAQs

• Q: What is 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.
• Q: Why is the domain important? A: The domain is important because it determines the set of input values for which the function is defined.
• Q: How do I determine the domain of a function? A: To determine the domain of a function, you need to consider the constraints on the input values, such as the presence of any restrictions or limitations.