To represent the situation where an inspector decides to stop production if the discrepancy in the diameter of a golf ball is more than [tex]\(0.002 \text{ mm}\)[/tex], we need a function that measures the absolute difference between the measured diameter [tex]\(x\)[/tex] and the standard diameter, which is [tex]\(42.67 \text{ mm}\)[/tex].
The absolute difference can be represented mathematically as:
[tex]\[ f(x) = |42.67 - x| \][/tex]
Let's break down why this function is appropriate:
1. Absolute Value: The absolute value ensures that the result is always a non-negative number, reflecting the size of the discrepancy irrespective of whether the measured diameter [tex]\(x\)[/tex] is larger or smaller than [tex]\(42.67 \text{ mm}\)[/tex].
2. Difference in Diameters: The expression within the absolute value, [tex]\(42.67 - x\)[/tex], represents the direct difference between the standard diameter (42.67 mm) and the measured diameter [tex]\(x\)[/tex].
Given the choices:
- [tex]\[ f(x) = x - |42.67| \][/tex]: This does not account for the absolute difference between the specific measured diameter [tex]\(x\)[/tex] and the standard diameter. Also, [tex]\(|42.67|\)[/tex] is just [tex]\(42.67\)[/tex] as it is a positive constant, so this bears no relation to the measurement [tex]\(x\)[/tex].
- [tex]\[ f(x) = |x| - 42.67 \][/tex]: This expression does not give the absolute difference directly. It first takes the modulus of [tex]\(x\)[/tex] which can distort the intention behind calculating discrepancy.
- [tex]\[ f(x) = |42.67 - x| \][/tex]: This correctly measures the absolute difference between the measured diameter and the standard diameter.
Hence, the correct function to represent the situation of measuring the discrepancy is:
[tex]\[ f(x) = |42.67 - x| \][/tex]
This function matches the requirements of measuring the absolute discrepancy between the measured diameter and the standard diameter. Therefore, we use [tex]\( f(x) = |42.67 - x| \)[/tex].
