Complete The Table.${ \begin{tabular}{|c|c|} \hline N N N & F ( N ) = − 4 ∣ N ∣ + 6 F(n) = -4|n| + 6 F ( N ) = − 4∣ N ∣ + 6 \ \hline -2 & □ \square □ \ \hline -1 & □ \square □ \ \hline 0 & □ \square □ \ \hline 1 & □ \square □ \ \hline \end{tabular} }$

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Introduction

In mathematics, tables are often used to represent functions and their corresponding values. A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. In this article, we will explore a given table that represents a function f(n)=−4∣n∣+6f(n) = -4|n| + 6 and complete the table with the missing values.

The Function f(n)=−4∣n∣+6f(n) = -4|n| + 6

The given function is defined as f(n)=−4∣n∣+6f(n) = -4|n| + 6, where nn is the input and f(n)f(n) is the output. The absolute value function ∣n∣|n| is used to ensure that the output is always non-negative, even if the input is negative.

To understand the behavior of this function, let's analyze its components:

  • The term −4∣n∣-4|n| represents a linear function that decreases as nn increases. The coefficient −4-4 indicates that the function decreases at a rate of 44 units for every unit increase in nn.
  • The term +6+6 represents a constant shift of 66 units to the right.

Completing the Table

The given table has five rows, each representing a different value of nn. The table is incomplete, with a placeholder □\square in each row. To complete the table, we need to calculate the corresponding value of f(n)f(n) for each row.

Calculating f(−2)f(-2)

For n=−2n = -2, we substitute the value into the function:

f(−2)=−4∣−2∣+6f(-2) = -4|-2| + 6 f(−2)=−4(2)+6f(-2) = -4(2) + 6 f(−2)=−8+6f(-2) = -8 + 6 f(−2)=−2f(-2) = -2

So, the value of f(−2)f(-2) is −2-2.

Calculating f(−1)f(-1)

For n=−1n = -1, we substitute the value into the function:

f(−1)=−4∣−1∣+6f(-1) = -4|-1| + 6 f(−1)=−4(1)+6f(-1) = -4(1) + 6 f(−1)=−4+6f(-1) = -4 + 6 f(−1)=2f(-1) = 2

So, the value of f(−1)f(-1) is 22.

Calculating f(0)f(0)

For n=0n = 0, we substitute the value into the function:

f(0)=−4∣0∣+6f(0) = -4|0| + 6 f(0)=−4(0)+6f(0) = -4(0) + 6 f(0)=0+6f(0) = 0 + 6 f(0)=6f(0) = 6

So, the value of f(0)f(0) is 66.

Calculating f(1)f(1)

For n=1n = 1, we substitute the value into the function:

f(1)=−4∣1∣+6f(1) = -4|1| + 6 f(1)=−4(1)+6f(1) = -4(1) + 6 f(1)=−4+6f(1) = -4 + 6 f(1)=2f(1) = 2

So, the value of f(1)f(1) is 22.

The Completed Table

Here is the completed table with the missing values filled in:

nn $f(n) = -4 n + 6$
-2 -2
-1 2
0 6
1 2

Conclusion

In this article, we explored a given table that represents a function f(n)=−4∣n∣+6f(n) = -4|n| + 6 and completed the table with the missing values. We analyzed the behavior of the function and calculated the corresponding values of f(n)f(n) for each row. The completed table provides a clear understanding of the function's behavior and its corresponding values.

Key Takeaways

  • The function f(n)=−4∣n∣+6f(n) = -4|n| + 6 represents a linear function that decreases as nn increases.
  • The absolute value function ∣n∣|n| ensures that the output is always non-negative, even if the input is negative.
  • The completed table provides a clear understanding of the function's behavior and its corresponding values.

Further Exploration

This article provides a basic understanding of the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 and its corresponding values. For further exploration, you can try the following:

  • Analyze the function's behavior for different values of nn.
  • Explore the relationship between the function and its corresponding values.
  • Use the function to model real-world scenarios, such as population growth or financial transactions.
    Frequently Asked Questions: Understanding the Function f(n)=−4∣n∣+6f(n) = -4|n| + 6 ====================================================================

Introduction

In our previous article, we explored the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 and completed the table with the missing values. In this article, we will address some of the most frequently asked questions about this function.

Q: What is the domain of the function f(n)=−4∣n∣+6f(n) = -4|n| + 6?

A: The domain of the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 is all real numbers, denoted as R\mathbb{R}. This means that the function can take any real value as input.

Q: What is the range of the function f(n)=−4∣n∣+6f(n) = -4|n| + 6?

A: The range of the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 is all non-negative real numbers, denoted as [0,∞)[0, \infty). This means that the function can only output non-negative values.

Q: How does the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 behave for negative values of nn?

A: For negative values of nn, the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 behaves as a linear function that decreases as nn increases. This is because the absolute value function ∣n∣|n| ensures that the output is always non-negative, even if the input is negative.

Q: How does the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 behave for positive values of nn?

A: For positive values of nn, the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 behaves as a linear function that decreases as nn increases. This is because the absolute value function ∣n∣|n| ensures that the output is always non-negative, even if the input is positive.

Q: Can the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 be used to model real-world scenarios?

A: Yes, the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 can be used to model real-world scenarios, such as population growth or financial transactions. For example, the function can be used to model the growth of a population over time, where the input nn represents the time period and the output f(n)f(n) represents the population size.

Q: How can the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 be used in optimization problems?

A: The function f(n)=−4∣n∣+6f(n) = -4|n| + 6 can be used in optimization problems, such as minimizing or maximizing the output of the function. For example, the function can be used to minimize the cost of a project, where the input nn represents the number of resources used and the output f(n)f(n) represents the cost.

Q: Can the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 be used in machine learning algorithms?

A: Yes, the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 can be used in machine learning algorithms, such as regression or classification. For example, the function can be used as a feature extractor to transform the input data into a more meaningful representation.

Conclusion

In this article, we addressed some of the most frequently asked questions about the function f(n)=−4∣n∣+6f(n) = -4|n| + 6. We hope that this article has provided a better understanding of the function and its applications.

Key Takeaways

  • The function f(n)=−4∣n∣+6f(n) = -4|n| + 6 has a domain of all real numbers and a range of all non-negative real numbers.
  • The function behaves as a linear function that decreases as nn increases for both negative and positive values of nn.
  • The function can be used to model real-world scenarios, such as population growth or financial transactions.
  • The function can be used in optimization problems, such as minimizing or maximizing the output of the function.
  • The function can be used in machine learning algorithms, such as regression or classification.

Further Exploration

This article provides a basic understanding of the function f(n)=−4∣n∣+6f(n) = -4|n| + 6 and its applications. For further exploration, you can try the following:

  • Analyze the function's behavior for different values of nn.
  • Explore the relationship between the function and its corresponding values.
  • Use the function to model real-world scenarios, such as population growth or financial transactions.
  • Use the function in optimization problems, such as minimizing or maximizing the output of the function.
  • Use the function in machine learning algorithms, such as regression or classification.