To determine which function represents the discrepancy in the diameter of the golf ball as described, let's break down the problem step-by-step:
1. Understanding the Problem:
- The standard diameter of the golf ball is [tex]\(42.67 \, \text{mm}\)[/tex].
- The discrepancy we're concerned about is the difference between the measured diameter [tex]\(x\)[/tex] of a golf ball and the standard diameter [tex]\(42.67 \, \text{mm}\)[/tex].
- We are looking for a function that captures this difference, expressed as the absolute value of the difference since discrepancies can be both positive and negative.
2. Formulating the Discrepancy:
- The key element here is the absolute difference between [tex]\(x\)[/tex] and [tex]\(42.67\)[/tex].
- This absolute difference can be mathematically represented as [tex]\(|42.67 - x|\)[/tex].
3. Evaluating Given Functions:
- [tex]\(f(x) = x - |42.67|\)[/tex]:
This implies subtracting the absolute value of [tex]\(42.67\)[/tex] (which remains [tex]\(42.67\)[/tex]) from [tex]\(x\)[/tex]. This does not account for the absolute difference but rather a simple subtraction operation.
- [tex]\(f(x) = |x| - 42.67\)[/tex]:
This takes the absolute value of the measurement [tex]\(x\)[/tex] and subtracts [tex]\(42.67\)[/tex]. This function does not correctly represent the absolute difference between [tex]\(x\)[/tex] and [tex]\(42.67\)[/tex].
- [tex]\(f(x) = |42.67 - x|\)[/tex]:
This is the absolute value of the difference between [tex]\(42.67\)[/tex] and [tex]\(x\)[/tex], which perfectly captures the discrepancy between the measured diameter and the standard diameter.
- [tex]\(f(x) = 42.67 - |x|\)[/tex]:
This incorrectly takes the absolute value of [tex]\(x\)[/tex] first and then subtracts it from [tex]\(42.67\)[/tex], which does not accurately reflect the requirement for the absolute difference.
4. Conclusion:
Based on the requirement to accurately capture the discrepancy between the measured diameter and the standard diameter of [tex]\(42.67 \, \text{mm}\)[/tex]:
The appropriate function that represents this situation is:
[tex]\[
f(x) = |42.67 - x|
\][/tex]
This function properly measures the absolute discrepancy in the diameter of the golf balls, ensuring quality control can determine if the difference exceeds the allowable [tex]\(0.002 \, \text{mm}\)[/tex].
