To determine which function represents the situation of measuring the discrepancy in diameter of golf balls, let's carefully analyze each function.
Given:
- The standard diameter of a golf ball is 42.67 mm.
- The acceptable discrepancy in the diameter is 0.002 mm.
The function we need must indicate the deviation from the standard diameter 42.67 mm. Specifically, it should measure how far the measured diameter deviates from this standard.
### Analyzing the Functions:
1. Function 1: [tex]\( f(x) = x - |42.67| \)[/tex]
- Here, [tex]\( |42.67| \)[/tex] is simply 42.67 because the absolute value of a positive number is the number itself.
- This simplifies to [tex]\( f(x) = x - 42.67 \)[/tex].
- This function does not measure the deviation; instead, it subtracts a constant value (42.67), which does not properly account for the absolute value of the deviation.
2. Function 2: [tex]\( f(x) = |x| - 42.67 \)[/tex]
- This takes the absolute value of [tex]\( x \)[/tex] and then subtracts 42.67.
- It does not work because it assumes [tex]\( x \)[/tex] could be negative initially, but a diameter (as a length) is always positive.
- Moreover, it does not reflect the absolute deviation from the standard diameter.
3. Function 3: [tex]\( f(x) = |42.67 - x| \)[/tex]
- This function finds the absolute value of the difference between 42.67 and [tex]\( x \)[/tex].
- It directly represents the absolute deviation from the standard diameter.
- If [tex]\( x \)[/tex] is the measured diameter, then [tex]\( |42.67 - x| \)[/tex] gives the magnitude of the discrepancy without considering the direction.
- This reflects the correct method of measuring how far [tex]\( x \)[/tex] deviates from 42.67, regardless of whether [tex]\( x \)[/tex] is greater or less than 42.67.
4. Function 4: [tex]\( f(x) = 42.67 - |x| \)[/tex]
- Here, [tex]\( |x| \)[/tex] is the absolute value of the measurement, which doesn't make sense for a diameter as it is by nature positive.
- This function subtracts the absolute value of [tex]\( x \)[/tex] from 42.67, which is not what is needed for measuring the deviation from the standard diameter.
- This would give a negative result when [tex]\( x \)[/tex] (which is always positive) is measured against 42.67, making it unsuitable for representing the deviation.
### Conclusion:
The correct function to represent the situation where we need to measure how much a measured diameter deviates from a standard diameter of 42.67 mm is:
[tex]\[ f(x) = |42.67 - x| \][/tex]
This function accurately reflects the absolute deviation from the standard diameter.
Hence, the correct function is:
[tex]\[ f(x) = |42.67 - x| \][/tex]
And the corresponding index of this function in the provided list is:
[tex]\[ \boxed{3} \][/tex]
