1. TRANSLATE the constraint equation
- Given constraint: \(|\mathrm{f(x) - 17}| = 0\) for all real numbers x
- This tells us: The absolute value of \(\mathrm{(f(x) - 17)}\) must equal zero
2. INFER what this constraint means mathematically
- Key insight: An absolute value equals zero only when the expression inside equals zero
- Therefore: \(\mathrm{f(x) - 17 = 0}\)
- This gives us: \(\mathrm{f(x) = 17}\)
3. INFER the type of function this creates
- Since \(\mathrm{f(x) = 17}\) for ALL values of x, this is a constant function
- Every input produces the same output: 17
4. APPLY CONSTRAINTS to check each table
- The constraint says "for all real numbers x," so EVERY entry must show \(\mathrm{f(x) = 17}\)
- Choice A: Shows \(\mathrm{f(x) = 0}\) → doesn't match \(\mathrm{f(x) = 17}\)
- Choice B: Shows \(\mathrm{f(x) = 17}\) for all x values → matches our requirement ✓
- Choice C: Shows varying values (0, 17, 34) → violates constant function requirement
- Choice D: Shows varying values (17, 0, -17) → violates constant function requirement
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may misinterpret \(|\mathrm{f(x) - 17}| = 0\) as meaning "f(x) can be any value as long as it's 17 units away from zero" or get confused about absolute value properties.
This leads them to think f(x) could equal 17 or -17, making them select Choice D (17, 0, -17) because it contains the "target" value 17.
Second Most Common Error:
Inadequate APPLY CONSTRAINTS reasoning: Students correctly determine that \(\mathrm{f(x) = 17}\), but don't fully process that the constraint applies "for all real numbers x," meaning EVERY table entry must be 17.
They might focus only on seeing the value 17 somewhere in the table, leading them to select Choice C (0, 17, 34) because it contains 17, even though the other values violate the constraint.
The Bottom Line:
This problem tests understanding of absolute value properties combined with function behavior. Success requires both translating the absolute value constraint correctly AND recognizing that "for all x" means the function must be constant.
