Select The Correct Answer.Consider This Absolute Value Function: F ( X ) = ∣ X + 3 ∣ F(x) = |x + 3| F ( X ) = ∣ X + 3∣ If Function F F F Is Written As A Piecewise Function, Which Piece Will It Include?A. X + 3 , X ≥ 3 X + 3, \, X \geq 3 X + 3 , X ≥ 3 B. X + 3 , X ≥ − 3 X + 3, \, X \geq -3 X + 3 , X ≥ − 3 C.

Introduction

Absolute value functions are a fundamental concept in mathematics, particularly in algebra and calculus. They are used to represent the distance of a number from zero on the number line. In this article, we will explore the concept of absolute value functions and how they can be represented as piecewise functions. We will also examine a specific absolute value function, $f(x) = |x + 3|$, and determine which piece it will include when written as a piecewise function.

What is an Absolute Value Function?

An absolute value function is a function that takes a real number as input and returns its distance from zero on the number line. The absolute value of a number $x$ 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 other words, the absolute value of a number is its value without regard to its sign. For example, the absolute value of $-3$ is $3$, and the absolute value of $4$ is $4$.

Representing Absolute Value Functions as Piecewise Functions

A piecewise function is a function that is defined by multiple sub-functions, each of which is defined on a specific interval. In the case of absolute value functions, we can represent them as piecewise functions by considering the two cases: when the input is non-negative and when the input is negative.

For the absolute value function $f(x) = |x + 3|$, we can represent it as a piecewise function as follows:

$$f(x) = \begin{cases} x + 3, & \text{if } x \geq -3 \\ -(x + 3), & \text{if } x < -3 \end{cases}$$

Analyzing the Given Function

Now, let's analyze the given function $f(x) = |x + 3|$. We want to determine which piece it will include when written as a piecewise function.

To do this, we need to consider the two cases: when $x \geq -3$ and when $x < -3$.

Case 1: $x \geq -3$

When $x \geq -3$, the expression $x + 3$ is non-negative. Therefore, the absolute value function $f(x) = |x + 3|$ simplifies to $x + 3$.

Case 2: $x < -3$

When $x < -3$, the expression $x + 3$ is negative. Therefore, the absolute value function $f(x) = |x + 3|$ simplifies to $-(x + 3)$.

Conclusion

Based on our analysis, we can conclude that the piecewise function for $f(x) = |x + 3|$ will include the following pieces:

• $x + 3, \, x \geq -3$
• $-(x + 3), \, x < -3$

Therefore, the correct answer is:

A. $x + 3, \, x \geq -3$

Final Thoughts

In this article, we explored the concept of absolute value functions and how they can be represented as piecewise functions. We analyzed a specific absolute value function, $f(x) = |x + 3|$, and determined which piece it will include when written as a piecewise function. We hope that this article has provided a clear understanding of absolute value functions and how they can be represented in a piecewise form.

References

• [1] "Absolute Value Functions" by Math Open Reference
• [2] "Piecewise Functions" by Khan Academy

Discussion

Introduction

In our previous article, we explored the concept of absolute value functions and how they can be represented as piecewise functions. We analyzed a specific absolute value function, $f(x) = |x + 3|$, and determined which piece it will include when written as a piecewise function. In this article, we will continue to delve deeper into the world of absolute value functions and answer some frequently asked questions.

Q&A

Q: What is the difference between an absolute value function and a piecewise function?

A: An absolute value function is a function that takes a real number as input and returns its distance from zero on the number line. A piecewise function, on the other hand, is a function that is defined by multiple sub-functions, each of which is defined on a specific interval.

Q: How do I determine which piece an absolute value function will include when written as a piecewise function?

A: To determine which piece an absolute value function will include, you need to consider the two cases: when the input is non-negative and when the input is negative. If the input is non-negative, the absolute value function will simplify to the input itself. If the input is negative, the absolute value function will simplify to the negative of the input.

Q: Can absolute value functions be used to model real-world scenarios?

A: Yes, absolute value functions can be used to model real-world scenarios. For example, the distance between two points on a number line can be represented by an absolute value function. Additionally, the absolute value function can be used to model the cost of a product that increases or decreases based on the quantity sold.

Q: How do I graph an absolute value function?

A: To graph an absolute value function, you need to consider the two cases: when the input is non-negative and when the input is negative. When the input is non-negative, the graph will be a straight line that passes through the origin. When the input is negative, the graph will be a straight line that passes through the origin, but with a negative slope.

Q: Can absolute value functions be used to solve equations?

A: Yes, absolute value functions can be used to solve equations. For example, the equation $|x + 3| = 5$ can be solved by considering the two cases: when $x + 3 \geq 0$ and when $x + 3 < 0$.

Q: How do I find the domain of an absolute value function?

A: To find the domain of an absolute value function, you need to consider the two cases: when the input is non-negative and when the input is negative. The domain of the absolute value function will be all real numbers except for the values that make the input negative.

Q: Can absolute value functions be used to model periodic phenomena?

A: Yes, absolute value functions can be used to model periodic phenomena. For example, the absolute value function can be used to model the temperature of a city that varies periodically throughout the year.

Conclusion

In this article, we have answered some frequently asked questions about absolute value functions. We have discussed the difference between absolute value functions and piecewise functions, how to determine which piece an absolute value function will include, and how to graph an absolute value function. We have also discussed how absolute value functions can be used to model real-world scenarios and solve equations. We hope that this article has provided a clear understanding of absolute value functions and how they can be used in a variety of contexts.

References

• [1] "Absolute Value Functions" by Math Open Reference
• [2] "Piecewise Functions" by Khan Academy
• [3] "Graphing Absolute Value Functions" by Math Is Fun

Discussion

What are some other examples of absolute value functions that can be used to model real-world scenarios? How do you think the concept of absolute value functions can be applied in different fields, such as physics, engineering, or economics? Share your thoughts and ideas in the comments below!