Complete The Table For The Function F ( J ) = ∣ J ∣ F(j) = |j| F ( J ) = ∣ J ∣ .${ \begin{tabular}{|c|c|} \hline I I I & F ( J ) F(j) F ( J ) \ \hline -2 & □ \square □ \ \hline 0 & □ \square □ \ \hline 2 & □ \square □ \ \hline 4 & I I I \ \hline \end{tabular} }$Submit Your Answers

Introduction

In mathematics, the absolute value function is a fundamental concept that plays a crucial role in various mathematical operations. The absolute value of a number is its distance from zero on the number line, without considering direction. In this article, we will focus on completing the table for the function $f(j) = |j|$. We will explore the concept of absolute value, understand the function $f(j) = |j|$, and complete the table with the correct values.

Understanding the Absolute Value Function

The absolute value function is denoted by $|x|$ and is defined as:

$$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$

In simpler terms, if the number is positive or zero, the absolute value is the number itself. If the number is negative, the absolute value is the number multiplied by -1.

The Function $f(j) = |j|$

The function $f(j) = |j|$ is a simple absolute value function where the input is $j$ and the output is the absolute value of $j$. To complete the table, we need to find the absolute value of each input value.

Completing the Table

$i$	$f(j)$
-2	$\square$
0	$\square$
2	$\square$
4	$I$

To complete the table, we need to find the absolute value of each input value.

• For $i = -2$, the absolute value is $|-2| = 2$.
• For $i = 0$, the absolute value is $|0| = 0$.
• For $i = 2$, the absolute value is $|2| = 2$.
• For $i = 4$, the absolute value is $|4| = 4$.

Filled-in Table

$i$	$f(j)$
-2	2
0	0
2	2
4	4

Conclusion

In this article, we completed the table for the function $f(j) = |j|$. We understood the concept of absolute value, defined the absolute value function, and applied it to the function $f(j) = |j|$. The completed table shows the absolute value of each input value.

Absolute Value Function Properties

The absolute value function has several important properties that are useful in various mathematical operations.

• Non-Negativity: The absolute value of any number is always non-negative.
• Identity of Indiscernibles: If $|x| = |y|$, then $x = y$ or $x = -y$.
• Triangle Inequality: $|x + y| \leq |x| + |y|$.

These properties make the absolute value function a fundamental tool in mathematics.

Applications of Absolute Value Function

The absolute value function has numerous applications in various fields, including:

• Mathematics: Absolute value is used in algebra, geometry, and calculus.
• Physics: Absolute value is used to describe the magnitude of vectors and forces.
• Engineering: Absolute value is used in signal processing and control systems.

In conclusion, the absolute value function is a fundamental concept in mathematics that has numerous applications in various fields. Understanding the absolute value function and its properties is essential for solving mathematical problems and applying mathematical concepts to real-world problems.

Final Thoughts

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Introduction

In our previous article, we completed the table for the function $f(j) = |j|$. We understood the concept of absolute value, defined the absolute value function, and applied it to the function $f(j) = |j|$. In this article, we will answer some frequently asked questions about the absolute value function.

Q&A

Q: What is the absolute value of a negative number?

A: The absolute value of a negative number is its distance from zero on the number line, without considering direction. For example, the absolute value of -3 is 3.

Q: How do I calculate the absolute value of a number?

A: To calculate the absolute value of a number, you can use the following formula:

$$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$

Q: What is the difference between absolute value and distance?

A: Absolute value and distance are related but not exactly the same thing. Distance refers to the length of a line segment between two points, while absolute value refers to the distance from zero on the number line.

Q: Can I use absolute value in real-world applications?

A: Yes, absolute value has numerous applications in various fields, including physics, engineering, and finance. For example, absolute value can be used to describe the magnitude of vectors and forces in physics, or to calculate the absolute return on investment in finance.

Q: How do I graph the absolute value function?

A: To graph the absolute value function, you can use the following steps:

1. Plot the points (0, 0) and (1, 1) on the coordinate plane.
2. Draw a line segment from (0, 0) to (1, 1).
3. Reflect the line segment across the x-axis to create the graph of the absolute value function.

Q: Can I use absolute value in calculus?

A: Yes, absolute value is used extensively in calculus, particularly in the study of limits, derivatives, and integrals.

Q: How do I calculate the absolute value of a complex number?

A: To calculate the absolute value of a complex number, you can use the following formula:

$$|a + bi| = \sqrt{a^2 + b^2}$$

Q: Can I use absolute value in statistics?

A: Yes, absolute value is used in statistics to calculate the absolute deviation from the mean.

Q: How do I calculate the absolute value of a matrix?

A: To calculate the absolute value of a matrix, you can use the following formula:

$$|A| = \sqrt{\text{trace}(A^T A)}$$

Q: Can I use absolute value in machine learning?

A: Yes, absolute value is used in machine learning to calculate the absolute error between predicted and actual values.

Conclusion

In this article, we answered some frequently asked questions about the absolute value function. We covered topics such as calculating absolute value, graphing the absolute value function, and using absolute value in real-world applications. We hope this article has been helpful in clarifying any doubts you may have had about the absolute value function.

Absolute Value Function Resources

For further reading and resources on the absolute value function, we recommend the following:

• Khan Academy: Absolute Value
• Mathway: Absolute Value
• Wolfram MathWorld: Absolute Value
• MIT OpenCourseWare: Absolute Value

We hope this article has been helpful in your understanding of the absolute value function. If you have any further questions or topics you would like to discuss, please feel free to ask.