Select the correct answer. Consider this absolute value function: [tex]f(x)=|x+3|[/tex]. If function [tex]f[/tex] is written as a piecewise function, which piece will it include?
A. [tex]-x+3, x\ \textless \ -3[/tex]
B. [tex]x+3, x \geq-3[/tex]
C. [tex]-x-3, x\ \textless \ 3[/tex]
D. [tex]x+3, x \geq 3[/tex]

The absolute value function f ( x ) = ∣ x + 3∣ is defined as x + 3 when x + 3 ≥ 0 and − ( x + 3 ) when x + 3 < 0 .
Simplify the conditions to get f ( x ) = x + 3 when x ≥ − 3 and f ( x ) = − x − 3 when x < − 3 .
Compare the pieces with the given options.
The correct answer is x + 3 , x ≥ − 3 ​ .

Explanation

Understanding the Problem We are given the absolute value function f ( x ) = ∣ x + 3∣ and asked to determine which piece of its piecewise representation is correct.

Definition of Absolute Value Recall that the absolute value of a number is defined as: ∣ x ∣ = { x , ​ if x ≥ 0 − x , ​ if x < 0 ​

Applying to the Given Function Applying this definition to f ( x ) = ∣ x + 3∣ , we have: f ( x ) = { x + 3 , ​ if x + 3 ≥ 0 − ( x + 3 ) , ​ if x + 3 < 0 ​

Simplifying the Conditions Simplifying the conditions, we get: f ( x ) = { x + 3 , ​ if x ≥ − 3 − x − 3 , ​ if x < − 3 ​

Comparing with the Options Now, let's examine the given options:


Option A: − x + 3 , x < − 3 . This is incorrect because when x < − 3 , f ( x ) = − x − 3 , not − x + 3 .
Option B: x + 3 , x ≥ − 3 . This is correct because when x ≥ − 3 , f ( x ) = x + 3 .
Option C: − x − 3 , x < 3 . This is incorrect because the condition should be x < − 3 , not x < 3 .
Option D: x + 3 , x ≥ 3 . This is incorrect because the condition should be x ≥ − 3 , not x ≥ 3 .

Conclusion Therefore, the correct answer is B.

Examples
Absolute value functions are used in many real-world applications, such as calculating distances or deviations from a target value. For example, in engineering, you might use an absolute value function to ensure that a manufactured part is within a certain tolerance range. If the target length of a part is 10 cm, and the tolerance is ± 0.1 cm, the actual length x must satisfy ∣ x − 10∣ ≤ 0.1 . This means the length can be between 9.9 cm and 10.1 cm. Understanding absolute value functions helps in quality control and ensuring precision in manufacturing.

Answered by GinnyAnswer | 2025-07-08

The absolute value function f ( x ) = ∣ x + 3∣ can be expressed as a piecewise function: f ( x ) = { x + 3 , − x − 3 , ​ x ≥ − 3 x < − 3 ​ . The correct option from the given choices is Option B: x + 3 , x ≥ − 3 .
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Answered by Anonymous | 2025-07-18