Discussion and Further Exploration

Oct 13, 2025 by leolavanderia.com.br 122 views

Introduction

In mathematics, functions are a fundamental concept that helps us describe relationships between variables. A function is a rule that assigns to each input value, or input, exactly one output value, or output. In this article, we will explore a specific function, denoted as $f(b) = |-4b|$ , and complete a table that represents the function's behavior for different input values of $b$ . We will delve into the world of mathematical functions, exploring the properties and behavior of this particular function.

The Function $f(b) = |-4b|$

The function $f(b) = |-4b|$ is a simple yet powerful example of a function that involves absolute value. The absolute value of a number is its distance from zero on the number line, without considering direction. In this case, the function takes the input value $b$ and multiplies it by $-4$ , resulting in a negative value. The absolute value of this negative value is then taken, effectively making it positive.

Completing the Table

The table provided is incomplete, with blank squares representing the function's output values for different input values of $b$ . Let's fill in the table by evaluating the function for each input value.

Evaluating $f(-2)$

To evaluate $f(-2)$ , we substitute $b = -2$ into the function:

$f(-2) = |-4(-2)|$

Using the properties of absolute value, we can simplify this expression:

$f(-2) = |8|$

Since the absolute value of $8$ is $8$ , we have:

$f(-2) = 8$

Evaluating $f(-1)$

Next, we evaluate $f(-1)$ by substituting $b = -1$ into the function:

$f(-1) = |-4(-1)|$

Simplifying this expression, we get:

$f(-1) = |4|$

The absolute value of $4$ is $4$ , so we have:

$f(-1) = 4$

Evaluating $f(0)$

Now, let's evaluate $f(0)$ by substituting $b = 0$ into the function:

$f(0) = |-4(0)|$

Since any number multiplied by $0$ is $0$ , we have:

$f(0) = |0|$

The absolute value of $0$ is $0$ , so we have:

$f(0) = 0$

Evaluating $f(1)$

Finally, we evaluate $f(1)$ by substituting $b = 1$ into the function:

$f(1) = |-4(1)|$

Simplifying this expression, we get:

$f(1) = |-4|$

The absolute value of $-4$ is $4$ , so we have:

$f(1) = 4$

Conclusion

In this article, we explored the function $f(b) = |-4b|$ and completed a table that represents its behavior for different input values of $b$ . We evaluated the function for each input value, using the properties of absolute value to simplify the expressions. The completed table provides a clear picture of the function's behavior, demonstrating how it maps input values to output values.

The Completed Table

$b$	$f(b)$
-2	8
-1	4
0	0
1	4

Discussion and Further Exploration

The function $f(b) = |-4b|$ is a simple yet powerful example of a function that involves absolute value. By exploring this function, we have gained a deeper understanding of the properties and behavior of absolute value functions. This knowledge can be applied to a wide range of mathematical contexts, from algebra to calculus.

In addition to exploring the function $f(b) = |-4b|$ , we can also consider other functions that involve absolute value. For example, we can investigate the function $g(x) = |x^2 - 4|$ , which involves the absolute value of a quadratic expression. By exploring these functions, we can gain a deeper understanding of the properties and behavior of absolute value functions, and develop a more nuanced appreciation for the mathematical concepts that underlie them.

References

[1] "Functions" by Khan Academy
[2] "Absolute Value" by Math Open Reference
[3] "Functions and Relations" by Wolfram MathWorld

Q: What is the function $f(b) = |-4b|$ ?

A: The function $f(b) = |-4b|$ is a simple function that involves absolute value. It takes the input value $b$ and multiplies it by $-4$ , resulting in a negative value. The absolute value of this negative value is then taken, effectively making it positive.

Q: How do I evaluate the function $f(b) = |-4b|$ for different input values of $b$ ?

A: To evaluate the function $f(b) = |-4b|$ for different input values of $b$ , you can substitute the input value into the function and simplify the expression using the properties of absolute value.

Q: What is the completed table for the function $f(b) = |-4b|$ ?

A: The completed table for the function $f(b) = |-4b|$ is:

$b$	$f(b)$
-2	8
-1	4
0	0
1	4
**Q: How do I understand the behavior of the function $f(b) =	-4b	$?**

A: To understand the behavior of the function $f(b) = |-4b|$ , you can analyze the completed table and observe how the function maps input values to output values. You can also consider the properties of absolute value and how they affect the function's behavior.

Q: What are some real-world applications of the function $f(b) = |-4b|$ ?

A: The function $f(b) = |-4b|$ has several real-world applications, including:

Modeling the behavior of physical systems that involve absolute value, such as the motion of an object under the influence of a force.
Analyzing the behavior of electrical circuits that involve absolute value, such as the voltage across a resistor.
Solving problems in physics, engineering, and other fields that involve absolute value.

Q: How do I extend the function $f(b) = |-4b|$ to other input values of $b$ ?

A: To extend the function $f(b) = |-4b|$ to other input values of $b$ , you can use the properties of absolute value to simplify the expression and evaluate the function for the new input value.

Q: What are some common mistakes to avoid when working with the function $f(b) = |-4b|$ ?

A: Some common mistakes to avoid when working with the function $f(b) = |-4b|$ include:

Failing to simplify the expression using the properties of absolute value.
Not considering the behavior of the function for different input values of $b$ .
Not using the completed table to understand the behavior of the function.

Q: How do I use the function $f(b) = |-4b|$ in a mathematical proof?

A: To use the function $f(b) = |-4b|$ in a mathematical proof, you can use the properties of absolute value to simplify the expression and evaluate the function for different input values of $b$ . You can also use the completed table to understand the behavior of the function and make conclusions about the proof.

Q: What are some advanced topics related to the function $f(b) = |-4b|$ ?

A: Some advanced topics related to the function $f(b) = |-4b|$ include:

Analyzing the behavior of the function for different input values of $b$ using calculus.
Using the function $f(b) = |-4b|$ to model complex systems that involve absolute value.
Developing new mathematical techniques and tools to analyze and solve problems involving absolute value.

Note: The questions and answers provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.